{"history":[],"article":{"id":"fb87f185-19bd-4352-9a61-b2871f8e90fb","title":"Logarithm","slug":"logarithm","content":":::summary\n:::\n\n# \n[[File:Logarithm plots.png|right|thumb|upright=1.35|Plots of logarithm functions, with three commonly used bases. The special points\nlogb b = 1\nare indicated by dotted lines, and all curves intersect in\nlogb 1 = 0\n.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the **logarithm** to ****base**** b is the inverse function of [exponentiation](https://bliptext.com/articles/[exponent](https://bliptext.com/articles/exponent)iation) with ****base**** b. That means that the **logarithm** of a number x to the ****base**** b is the [exponent](https://bliptext.com/articles/exponent) to which b must be raised to produce x. For example, since\n, the logarithm base 10 of\n1000\nis\n3\n, or\nlog10 (1000) = 3\n. The logarithm of x to base b is denoted as\nlogb (x)\n, or without parentheses,\nlogb x\n. When the base is clear from the context or is irrelevant it is sometimes written\nlog x\n.\nThe logarithm base\n10\nis called the decimal or common logarithm and is commonly used in science and engineering.\nThe natural logarithm has the number [[e (mathematical constant)|\ne ≈ 2.718\n]] as its base; its use is widespread in mathematics and [physics](https://bliptext.com/articles/physics) because of its very simple [derivative](https://bliptext.com/articles/derivative).\nThe binary logarithm uses base\n2\nand is frequently used in [computer science](https://bliptext.com/articles/computer-science).\nLogarithms were introduced by [John Napier](https://bliptext.com/articles/john-napier) in 1614 as a means of simplifying calculations.\nThey were rapidly adopted by [navigators](https://bliptext.com/articles/navigator), scientists, engineers, [surveyors](https://bliptext.com/articles/surveying), and others to perform high-accuracy computations more easily.\nUsing [logarithm tables](https://bliptext.com/articles/logarithm-table), tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition.\nThis is possible because the logarithm of a [product](https://bliptext.com/articles/product-mathematics) is the [sum](https://bliptext.com/articles/summation) of the logarithms of the factors:\nprovided that b, x and y are all positive and\nb ≠ 1\n. The [slide rule](https://bliptext.com/articles/slide-rule), also based on logarithms, allows quick calculations without tables, but at lower precision.\nThe present-day notion of logarithms comes from [Leonhard Euler](https://bliptext.com/articles/leonhard-euler), who connected them to the [exponential function](https://bliptext.com/articles/exponential-function) in the 18th century, and who also introduced the letter e as the base of natural logarithms.\n[Logarithmic scales](https://bliptext.com/articles/logarithmic-scale) reduce wide-ranging quantities to smaller scopes.\nFor example, the [decibel](https://bliptext.com/articles/decibel) (dB) is a [unit](https://bliptext.com/articles/units-of-measurement) used to express [ratio as logarithms](https://bliptext.com/articles/level-logarithmic-quantity), mostly for signal power and amplitude (of which [sound pressure](https://bliptext.com/articles/sound-pressure) is a common example).\nIn chemistry, [pH](https://bliptext.com/articles/ph) is a logarithmic measure for the [acidity](https://bliptext.com/articles/acid) of an [aqueous solution](https://bliptext.com/articles/aqueous-solution).\nLogarithms are commonplace in scientific [formulae](https://bliptext.com/articles/formula), and in measurements of the [complexity of algorithms](https://bliptext.com/articles/computational-complexity-theory) and of geometric objects called [fractals](https://bliptext.com/articles/fractal).\nThey help to describe [frequency](https://bliptext.com/articles/frequency) ratios of [musical intervals](https://bliptext.com/articles/interval-music), appear in formulas counting [prime numbers](https://bliptext.com/articles/prime-number) or [approximating](https://bliptext.com/articles/stirling-s-approximation) [factorials](https://bliptext.com/articles/factorial), inform some models in [psychophysics](https://bliptext.com/articles/psychophysics), and can aid in [forensic accounting](https://bliptext.com/articles/forensic-accounting).\nThe concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well.\nHowever, in general settings, the logarithm tends to be a multi-valued function.\nFor example, the [complex logarithm](https://bliptext.com/articles/complex-logarithm) is the multi-valued [inverse](https://bliptext.com/articles/inverse-function) of the complex exponential function.\nSimilarly, the [discrete logarithm](https://bliptext.com/articles/discrete-logarithm) is the multi-valued inverse of the exponential function in finite groups; it has uses in [public-key cryptography](https://bliptext.com/articles/public-key-cryptography).\n\n# Motivation\nBinary logarithm plot with grid.png of the logarithm base 2 crosses the x-axis at\nand passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g.,\nand\n. The graph gets arbitrarily close to the y-axis, but [does not meet it](https://bliptext.com/articles/asymptotic).]]\n[Addition](https://bliptext.com/articles/addition), [multiplication](https://bliptext.com/articles/multiplication), and [exponentiation](https://bliptext.com/articles/exponentiation) are three of the most fundamental arithmetic operations.\nThe inverse of addition is [subtraction](https://bliptext.com/articles/subtraction), and the inverse of multiplication is [division](https://bliptext.com/articles/division-mathematics).\nSimilarly, a logarithm is the inverse operation of [exponentiation](https://bliptext.com/articles/exponentiation).\nExponentiation is when a number b, the base, is raised to a certain power y, the exponent, to give a value x; this is denoted\nb^y=x.\nFor example, raising\n2\nto the power of\n3\ngives\n8\nThe logarithm of base b is the inverse operation, that provides the output y from the input x. That is, is equivalent to x=b^y if b is a positive [real number](https://bliptext.com/articles/real-number).\n(If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)\nOne of the main historical motivations of introducing logarithms is the formula\nby which [tables of logarithms](https://bliptext.com/articles/logarithm-table) allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.\n\n# Definition\nGiven a positive [real number](https://bliptext.com/articles/real-number) b such that\nb ≠ 1\n, the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that b^y = x.\nThe logarithm is denoted \"\nlogb x\n\" (pronounced as \"the logarithm of x to base b\", \"the base-b logarithm of x\", or most commonly \"the log, base b, of x\").\nAn equivalent and more succinct definition is that the function\nlogb\nis the [inverse function](https://bliptext.com/articles/inverse-function) to the function.\n\n## Examples\nlog2 16 = 4\n, since\n24 = 2 × 2 × 2 × 2 = 16\n.\nlog10 150\nis approximately 2.176, which lies between 2 and 3, just as 150 lies between\n102 = 100\nand\n103 = 1000\n.\nlogb b = 1\nand\nlogb 1 = 0\n, since\nb1 = b\nand\nb0 = 1\n, respectively.\n\n# Logarithmic identities\nSeveral important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.\n\n## Product, quotient, power, and root\nThe logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms.\nThe logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples.\nEach of the identities can be derived after substitution of the logarithm definitions or in the left hand sides.\n\n## Change of base\nThe logarithm\nlogb x\ncan be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:\nTypical [scientific calculators](https://bliptext.com/articles/scientific-calculators) calculate the logarithms to bases 10 and e. Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:\nGiven a number x and its logarithm\ny = logb x\nto an unknown base b, the base is given by:\nwhich can be seen from taking the defining equation to the power of\n\n# Particular bases\nAmong all choices for the base, three are particularly common.\nThese are\n,\n(the [irrational](https://bliptext.com/articles/irrational-number) mathematical constant e ≈ 2.71828183\n), and\n(the [binary logarithm](https://bliptext.com/articles/binary-logarithm)).\nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), the logarithm base e is widespread because of analytical properties explained below.\nOn the other hand, base 10 logarithms (the [common logarithm](https://bliptext.com/articles/common-logarithm)) are easy to use for manual calculations in the [decimal](https://bliptext.com/articles/decimal) number system:\nThus,\nlog10 (x)\nis related to the number of [decimal digits](https://bliptext.com/articles/decimal-digit) of a positive [integer](https://bliptext.com/articles/integer) x: The number of digits is the smallest [integer](https://bliptext.com/articles/integer) strictly bigger than log10 (x)\n. For example,\nlog10(5986)\nis approximately 3.78.\nThe [next integer above](https://bliptext.com/articles/floor-and-ceiling-functions) it is 4, which is the number of digits of 5986.\nBoth the natural logarithm and the binary logarithm are used in [information theory](https://bliptext.com/articles/information-theory), corresponding to the use of [nats](https://bliptext.com/articles/nat-unit) or [bits](https://bliptext.com/articles/bit) as the fundamental units of information, respectively.\nBinary logarithms are also used in [computer science](https://bliptext.com/articles/computer-science), where the [binary system](https://bliptext.com/articles/binary-numeral-system) is ubiquitous; in [music theory](https://bliptext.com/articles/music-theory), where a pitch ratio of two (the [octave](https://bliptext.com/articles/octave)) is ubiquitous and the number of [cents](https://bliptext.com/articles/cent-music) between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 [cents](https://bliptext.com/articles/cent-music) per [semitone](https://bliptext.com/articles/semitone) in [conventional equal temperament](https://bliptext.com/articles/12-tone-equal-temperament)), or equivalently the log base 21/1200\n; and in [photography](https://bliptext.com/articles/photography) rescaled base 2 logarithms are used to measure [exposure values](https://bliptext.com/articles/exposure-value), [light levels](https://bliptext.com/articles/luminance), [exposure times](https://bliptext.com/articles/exposure-time), lens [apertures](https://bliptext.com/articles/aperture), and [film speeds](https://bliptext.com/articles/film-speed) in \"stops\".\nThe abbreviation\nlog x\nis often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial.\nCommon logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts\nlog x\nstill often means the base ten logarithm.\nIn mathematics\nlog x\nusually means to the natural logarithm (base e).\nIn computer science and information theory,\nlog\noften refers to binary logarithms (base 2).\nThe following table lists common notations for logarithms to these bases.\nThe \"ISO notation\" column lists designations suggested by the [International Organization for Standardization](https://bliptext.com/articles/international-organization-for-standardization).\n# [Overlaid graphs of the logarithm for bases 1⁄2, 2, and e | upload.wikimedia.org/wikipedia/commons/a/ae/Log4.svg]\n\n# History\nThe history of logarithms in seventeenth-century Europe saw the discovery of a new [function](https://bliptext.com/articles/function-mathematics) that extended the realm of analysis beyond the scope of algebraic methods.\nThe method of logarithms was publicly propounded by [John Napier](https://bliptext.com/articles/john-napier) in 1614, in a book titled [Mirifici Logarithmorum Canonis Descriptio](https://bliptext.com/articles/mirifici-logarithmorum-canonis-descriptio) (Description of the Wonderful Canon of Logarithms).\nPrior to Napier's invention, there had been other techniques of similar scopes, such as the [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis) or the use of tables of progressions, extensively developed by [Jost Bürgi](https://bliptext.com/articles/jost-b-rgi) around 1600.\nNapier coined the term for logarithm in Middle Latin, logarithmus, literally meaning ratio-number, derived from the Greek logos proportion, ratio, word + arithmos number.\nThe [common logarithm](https://bliptext.com/articles/common-logarithm) of a number is the index of that power of ten which equals the number.\nSpeaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by [Archimedes](https://bliptext.com/articles/archimedes) as the \"order of a number\".\nThe first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation.\nSome of these methods used tables derived from trigonometric identities.\nSuch methods are called [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis).\nInvention of the [function](https://bliptext.com/articles/function-mathematics) now known as the [natural logarithm](https://bliptext.com/articles/natural-logarithm) began as an attempt to perform a [quadrature](https://bliptext.com/articles/quadrature-mathematics) of a rectangular [hyperbola](https://bliptext.com/articles/hyperbola) by [Grégoire de Saint-Vincent](https://bliptext.com/articles/gr-goire-de-saint-vincent), a Belgian Jesuit residing in Prague.\nArchimedes had written [The Quadrature of the Parabola](https://bliptext.com/articles/the-quadrature-of-the-parabola) in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647.\nThe relation that the logarithm provides between a [geometric progression](https://bliptext.com/articles/geometric-progression) in its [argument](https://bliptext.com/articles/argument-of-a-function) and an [arithmetic progression](https://bliptext.com/articles/arithmetic-progression) of values, prompted [A. A. de Sarasa](https://bliptext.com/articles/a-a-de-sarasa) to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis), leading to the term \"hyperbolic logarithm\", a synonym for natural logarithm.\nSoon the new function was appreciated by [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens), and [James Gregory](https://bliptext.com/articles/james-gregory-mathematician).\nThe notation\nLog y\nwas adopted by [Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz) in 1675, and the next year he connected it to the [integral](https://bliptext.com/articles/integral-calculus)\nBefore Euler developed his modern conception of complex natural logarithms, [Roger Cotes](https://bliptext.com/articles/roger-cotes) had a nearly equivalent result when he showed in 1714 that\n\n# Logarithm tables, slide rules, and historical applications\nBy simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially [astronomy](https://bliptext.com/articles/astronomy).\nThey were critical to advances in [surveying](https://bliptext.com/articles/surveying), [celestial navigation](https://bliptext.com/articles/celestial-navigation), and other domains.\n[Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace) called logarithms\nAs the function\nis the inverse function of\nlogb x\n, it has been called an **antilogarithm**.\nNowadays, this function is more commonly called an [exponential function](https://bliptext.com/articles/exponential-function).\n# [The 1797 Encyclopædia Britannica explanation of logarithms | upload.wikimedia.org/wikipedia/commons/8/82/Logarithms///Britannica///1797.png]\n\n## Log tables\nA key tool that enabled the practical use of logarithms was the [table of logarithms](https://bliptext.com/articles/log-table).\nThe first such table was compiled by [Henry Briggs](https://bliptext.com/articles/henry-briggs-mathematician) in 1617, immediately after Napier's invention but with the innovation of using 10 as the base.\nBriggs' first table contained the [common logarithms](https://bliptext.com/articles/common-logarithm) of all integers in the range from 1 to 1000, with a precision of 14 digits.\nSubsequently, tables with increasing scope were written.\nThese tables listed the values of\nlog10 x\nfor any number x in a certain range, at a certain precision.\nBase-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers.\nThe common logarithm of x can be separated into an [integer part](https://bliptext.com/articles/integer-part) and a [fractional part](https://bliptext.com/articles/fractional-part), known as the characteristic and [mantissa](https://bliptext.com/articles/mantissa-logarithm).\nTables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.\nThe characteristic of\n10 · x\nis one plus the characteristic of x, and their mantissas are the same.\nThus using a three-digit log table, the logarithm of 3542 is approximated by\nGreater accuracy can be obtained by [interpolation](https://bliptext.com/articles/interpolation):\nThe value of\n10x\ncan be determined by reverse look up in the same table, since the logarithm is a [monotonic function](https://bliptext.com/articles/monotonic-function).\n\n## Computations\nThe product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms.\nThe product\ncd\nor quotient\nc/d\ncame from looking up the antilogarithm of the sum or difference, via the same table:\nand\nFor manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis), which relies on [trigonometric identities](https://bliptext.com/articles/trigonometric-identities).\nCalculations of powers and [roots](https://bliptext.com/articles/nth-root) are reduced to multiplications or divisions and lookups by\nand\nTrigonometric calculations were facilitated by tables that contained the common logarithms of [trigonometric functions](https://bliptext.com/articles/trigonometric-function).\n\n## Slide rules\nAnother critical application was the [slide rule](https://bliptext.com/articles/slide-rule), a pair of logarithmically divided scales used for calculation.\nThe non-sliding logarithmic scale, [Gunter's rule](https://bliptext.com/articles/gunter-s-rule), was invented shortly after Napier's invention.\n[William Oughtred](https://bliptext.com/articles/william-oughtred) enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other.\nNumbers are placed on sliding scales at distances proportional to the differences between their logarithms.\nSliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:\nFor example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part.\nThe slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.\n# [Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x. | upload.wikimedia.org/wikipedia/commons/8/8f/Slide///rule///example2///with///labels.svg]\n\n# Analytic properties\nA deeper study of logarithms requires the concept of a [function](https://bliptext.com/articles/function-mathematics).\nA function is a rule that, given one number, produces another number.\nAn example is the function producing the x-th power of b from any real number x, where the base b is a fixed number.\nThis function is written as\nf(x) = b x\n. When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals.\n\n## Existence\nLet b be a positive real number not equal to 1 and let\nf(x) = b x\n.\nIt is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range.\nThis fact follows from the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem).\nNow, f is [strictly increasing](https://bliptext.com/articles/monotonic-function) (for\nb > 1\n), or strictly decreasing (for\n0 < b < 1\n), is continuous, has domain \\R, and has range \\R_{> 0}. Therefore, f is a bijection from \\R to \\R_{>0}. In other words, for each positive real number y, there is exactly one real number x such that b^x = y.\nWe let denote the inverse of f. That is,\nlogb y\nis the unique real number x such that b^x = y. This function is called the base-b logarithm function or logarithmic function (or just logarithm).\n\n## Characterization by the product formula\nThe function\nlogb x\ncan also be essentially characterized by the product formula\nMore precisely, the logarithm to any base\nb > 1\nis the only [increasing function](https://bliptext.com/articles/increasing-function) f from the positive reals to the reals satisfying\nf(b) = 1\nand\n\n## Graph of the logarithm function\n[[File:Logarithm inversefunctiontoexp.svg|right|thumb|The graph of the logarithm function\nlogb (x)\n(blue) is obtained by [reflecting](https://bliptext.com/articles/reflection-mathematics) the graph of the function\nbx\n(red) at the diagonal line (\nx = y\n).|alt=The graphs of two functions.]] As discussed above, the function\nlogb\nis the inverse to the exponential function.\nTherefore, their [graphs](https://bliptext.com/articles/graph-of-a-function) correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line\nx = y\n), as shown at the right: a point\n(t, u = bt)\non the graph of f yields a point\n(u, t = logb u)\non the graph of the logarithm and vice versa.\nAs a consequence,\nlogb (x)\n[diverges to infinity](https://bliptext.com/articles/divergent-sequence) (gets bigger than any given number) if x grows to infinity, provided that b is greater than one.\nIn that case,\nlogb(x)\nis an [increasing function](https://bliptext.com/articles/increasing-function).\nFor\nb < 1\n,\nlogb (x)\ntends to minus infinity instead.\nWhen x approaches zero,\nlogb x\ngoes to minus infinity for\nb > 1\n(plus infinity for\nb < 1\n, respectively).\n\n## Derivative and antiderivative\nLogarithm derivative.svg (green) and its tangent at\n(black)|alt=A graph of the logarithm function and a line touching it in one point.]]\nAnalytic properties of functions pass to their inverses.\nThus, as\nf(x) = bx\nis a continuous and [differentiable function](https://bliptext.com/articles/differentiable-function), so is\nlogb y\n. Roughly, a continuous function is differentiable if its graph has no sharp \"corners\".\nMoreover, as the [derivative](https://bliptext.com/articles/derivative) of\nf(x)\nevaluates to\nln(b) bx\nby the properties of the [exponential function](https://bliptext.com/articles/exponential-function), the [chain rule](https://bliptext.com/articles/chain-rule) implies that the derivative of\nlogb x\nis given by\nThat is, the [slope](https://bliptext.com/articles/slope) of the [tangent](https://bliptext.com/articles/tangent) touching the graph of the\nbase-b\nlogarithm at the point\n(x, logb (x))\nequals\n1/(x ln(b))\n.\nThe derivative of\nln(x)\nis\n1/x\nln(x)\nis the unique [antiderivative](https://bliptext.com/articles/antiderivative) of\n1/x\nthat has the value 0 for\nx = 1\n. It is this very simple formula that motivated to qualify as \"natural\" the natural logarithm; this is also one of the main reasons of the importance of the [constant e](https://bliptext.com/articles/e-mathematical-constant).\nThe derivative with a generalized functional argument\nf(x)\nis\nThe quotient at the right hand side is called the [logarithmic derivative](https://bliptext.com/articles/logarithmic-derivative) of f.\nComputing\nf ' (x)\nby means of the derivative of\nln(f(x))\nis known as [logarithmic differentiation](https://bliptext.com/articles/logarithmic-differentiation).\nThe antiderivative of the [natural logarithm](https://bliptext.com/articles/natural-logarithm)\nln(x)\nis:\n[Related formulas](https://bliptext.com/articles/list-of-integrals-of-logarithmic-functions), such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.\n\n## Integral representation of the natural logarithm\nNatural logarithm integral.svg of t is the shaded area underneath the graph of the function\nf(x) = 1/x\n.|alt=A hyperbola with part of the area underneath shaded in grey.]]\nThe [natural logarithm](https://bliptext.com/articles/natural-logarithm) of t can be defined as the [definite integral](https://bliptext.com/articles/definite-integral):\nThis definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal.\nAs an integral,\nln(t)\nequals the area between the x-axis and the graph of the function\n1/x\n, ranging from\nx = 1\nto\nx = t\n. This is a consequence of the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) and the fact that the derivative of\nln(x)\nis\n1/x\n. Product and power logarithm formulas can be derived from this definition.\nFor example, the product formula\nln(tu) = ln(t) + ln(u)\nis deduced as:\nThe equality (1) splits the integral into two parts, while the equality (2) is a change of variable (\nw = x/t\n). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function\nf(x) = 1/x\nagain.\nTherefore, the left hand blue area, which is the integral of\nf(x)\nfrom t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof.\nThe power formula\nln(tr) = r ln(t)\nmay be derived in a similar way:\nThe second equality uses a change of variables ([integration by substitution](https://bliptext.com/articles/integration-by-substitution)),\nw = x1/r\n.\nThe sum over the reciprocals of natural numbers,\nis called the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics).\nIt is closely tied to the [natural logarithm](https://bliptext.com/articles/natural-logarithm): as n tends to [infinity](https://bliptext.com/articles/infinity), the difference,\n[converges](https://bliptext.com/articles/limit-of-a-sequence) (i.e. gets arbitrarily close) to a number known as the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant)\nγ = 0.5772...\n. This relation aids in analyzing the performance of algorithms such as [quicksort](https://bliptext.com/articles/quicksort).\n# [A visual proof of the product formula of the natural logarithm | upload.wikimedia.org/wikipedia/commons/9/9b/Natural///logarithm///product///formula///proven///geometrically.svg]\n\n## Transcendence of the logarithm\n[R[e](https://bliptext.com/articles/e-mathematical-constant)al numb[e](https://bliptext.com/articles/e-mathematical-constant)rs](https://blipt[e](https://bliptext.com/articles/e-mathematical-constant)xt.com/articl[e](https://bliptext.com/articles/e-mathematical-constant)s/r[e](https://bliptext.com/articles/e-mathematical-constant)al-numb[e](https://bliptext.com/articles/e-mathematical-constant)r) that ar[e](https://bliptext.com/articles/e-mathematical-constant) not [algebraic](https://blipt[e](https://bliptext.com/articles/e-mathematical-constant)xt.com/articl[e](https://bliptext.com/articles/e-mathematical-constant)s/alg[e](https://bliptext.com/articles/e-mathematical-constant)braic-numb[e](https://bliptext.com/articles/e-mathematical-constant)r) ar[e](https://bliptext.com/articles/e-mathematical-constant) call[e](https://bliptext.com/articles/e-mathematical-constant)d [transc[e](https://bliptext.com/articles/e-mathematical-constant)nd[e](https://bliptext.com/articles/e-mathematical-constant)ntal](https://blipt[e](https://bliptext.com/articles/e-mathematical-constant)xt.com/articl[e](https://bliptext.com/articles/e-mathematical-constant)s/transc[e](https://bliptext.com/articles/e-mathematical-constant)nd[e](https://bliptext.com/articles/e-mathematical-constant)ntal-numb[e](https://bliptext.com/articles/e-mathematical-constant)r); for [e](https://bliptext.com/articles/e-mathematical-constant)xampl[e](https://bliptext.com/articles/e-mathematical-constant), [π](https://blipt[e](https://bliptext.com/articles/e-mathematical-constant)xt.com/articl[e](https://bliptext.com/articles/e-mathematical-constant)s/pi) and [e](https://bliptext.com/articles/e-mathematical-constant) ar[e](https://bliptext.com/articles/e-mathematical-constant) such numb[e](https://bliptext.com/articles/e-mathematical-constant)rs, but is not.\n[Almost all](https://bliptext.com/articles/almost-all) real numbers are transcendental.\nThe logarithm is an example of a [transcendental function](https://bliptext.com/articles/transcendental-function).\nThe [Gelfond–Schneider theorem](https://bliptext.com/articles/gelfond-schneider-theorem) asserts that logarithms usually take transcendental, i.e. \"difficult\" values.\n\n# Calculation\nLogarithms are easy to compute in some cases, such as\nlog10 (1000) = 3\n. In general, logarithms can be calculated using [power series](https://bliptext.com/articles/power-series) or the [arithmetic–geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean), or be retrieved from a precalculated [logarithm table](https://bliptext.com/articles/logarithm-table) that provides a fixed precision.\n[Newton's method](https://bliptext.com/articles/newton-s-method), an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.\nUsing look-up tables, [CORDIC](https://bliptext.com/articles/cordic)-like methods can be used to compute logarithms by using only the operations of addition and [bit shifts](https://bliptext.com/articles/arithmetic-shift).\nMoreover, the [binary logarithm algorithm](https://bliptext.com/articles/binary-logarithm) calculates\nlb(x)\n[recursively](https://bliptext.com/articles/recursion), based on repeated squarings of x, taking advantage of the relation\n# [The logarithm keys (LOG for base 10 and LN for base e) on a TI-83 Plus graphing calculator | upload.wikimedia.org/wikipedia/commons/8/88/Logarithm///keys.jpg]\n\n## Power series\n\n### Taylor series\n[[File:Taylor approximation of natural logarithm.gif|right|thumb|The Taylor series of\nln(z)\ncentered at\n. The animation shows the first 10 approximations along with the 99th and 100th.\nThe approximations do not converge beyond a distance of 1 from the center.|alt=An animation showing increasingly good approximations of the logarithm graph.]]\nFor any real number z that satisfies\n0 < z ≤ 2\n, the following formula holds:\nEquating the function\nln(z)\nto this infinite sum ([series](https://bliptext.com/articles/series-mathematics)) is shorthand for saying that the function can be approximated to a more and more accurate value by the following expressions (known as [partial sums](https://bliptext.com/articles/partial-sum)):\nFor example, with\nthe third approximation yields\n0.4167\n, which is about\n0.011\ngreater than\n, and the ninth approximation yields\n0.40553\n, which is only about\n0.0001\ngreater.\nThe nth partial sum can approximate\nln(z)\nwith arbitrary precision, provided the number of summands n is large enough.\nIn elementary calculus, the series is said to [converge](https://bliptext.com/articles/convergent-series) to the function\nln(z)\n, and the function is the [limit](https://bliptext.com/articles/limit-mathematics) of the series.\nIt is the [Taylor series](https://bliptext.com/articles/taylor-series) of the [natural logarithm](https://bliptext.com/articles/natural-logarithm) at\nz = 1\n. The Taylor series of\nln(z)\nprovides a particularly useful approximation to\nln(1 + z)\nwhen z is small,\n, since then\nFor example, with\nz = 0.1\nthe first-order approximation gives\nln(1.1) ≈ 0.1\n, which is less than\n5%\noff the correct value\n0.0953\n.\n\n### Inverse hyperbolic tangent\nAnother series is based on the [inverse hyperbolic tangent](https://bliptext.com/articles/area-hyperbolic-tangent) function:\nfor any real number\nz > 0\n. Using [sigma notation](https://bliptext.com/articles/sigma-notation), this is also written as\nThis series can be derived from the above Taylor series.\nIt converges quicker than the Taylor series, especially if z is close to 1.\nFor example, for\nz = 1.5\n, the first three terms of the second series approximate\nln(1.5)\nwith an error of about 3.\nThe quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation\ny ≈ ln(z)\nand putting\nthe logarithm of z is:\nThe better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently.\nA can be calculated using the [exponential series](https://bliptext.com/articles/exponential-function), which converges quickly provided y is not too large.\nCalculating the logarithm of larger z can be reduced to smaller values of z by writing\n, so that\n.\nA closely related method can be used to compute the logarithm of integers.\nPutting in the above series, it follows that:\nIf the logarithm of a large integer n is known, then this series yields a fast converging series for\nlog(n+1)\n, with a [rate of convergence](https://bliptext.com/articles/rate-of-convergence) of.\n\n## Arithmetic–geometric mean approximation\nThe [arithmetic–geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean) yields high-precision approximations of the [natural logarithm](https://bliptext.com/articles/natural-logarithm).\nSasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed.\nIn their work\nln(x)\nis approximated to a precision of\n2−p\n(or p precise bits) by the following formula (due to [Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss)):\nHere\nM(x, y)\ndenotes the [arithmetic–geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean) of x and y. It is obtained by repeatedly calculating the average\n(x + y)/2\n([arithmetic mean](https://bliptext.com/articles/arithmetic-mean)) and \\sqrt{xy} ([geometric mean](https://bliptext.com/articles/geometric-mean)) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of\nM(x, y)\n. m is chosen such that\nto ensure the required precision.\nA larger m makes the\nM(x, y)\ncalculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision.\nThe constants\nπ\nand\nln(2)\ncan be calculated with quickly converging series.\n\n## Feynman's algorithm\nWhile at [Los Alamos National Laboratory](https://bliptext.com/articles/los-alamos-national-laboratory) working on the [Manhattan Project](https://bliptext.com/articles/manhattan-project), [Richard Feynman](https://bliptext.com/articles/richard-feynman) developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in the [Connection Machine](https://bliptext.com/articles/connection-machine).\nThe algorithm relies on the fact that every real number x where\n1 < x < 2\ncan be represented as a product of distinct factors of the form\n1 + 2−k\n. The algorithm sequentially builds that product P, starting with\nand\nP · (1 + 2−k) < x\n, then it changes P to\nP · (1 + 2−k)\n. It then increases k by one regardless.\nThe algorithm stops when k is large enough to give the desired accuracy.\nBecause\nlog(x)\nis the sum of the terms of the form\nlog(1 + 2−k)\ncorresponding to those k for which the factor\n1 + 2−k\nwas included in the product P,\nlog(x)\nmay be computed by simple addition, using a table of\nlog(1 + 2−k)\nfor all k. Any base may be used for the logarithm table.\n\n# Applications\nLogarithms have many applications inside and outside mathematics.\nSome of these occurrences are related to the notion of [scale invariance](https://bliptext.com/articles/scale-invariance).\nFor example, each chamber of the shell of a [nautilus](https://bliptext.com/articles/nautilus) is an approximate copy of the next one, scaled by a constant factor.\nThis gives rise to a [logarithmic spiral](https://bliptext.com/articles/logarithmic-spiral).\n[Benford's law](https://bliptext.com/articles/benford-s-law) on the distribution of leading digits can also be explained by scale invariance.\nLogarithms are also linked to [self-similarity](https://bliptext.com/articles/self-similarity).\nFor example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.\nThe dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.\n[Logarithmic scales](https://bliptext.com/articles/logarithmic-scale) are useful for quantifying the relative change of a value as opposed to its absolute difference.\nMoreover, because the logarithmic function\nlog(x)\ngrows very slowly for large x, logarithmic scales are used to compress large-scale scientific data.\nLogarithms also occur in numerous scientific formulas, such as the [Tsiolkovsky rocket equation](https://bliptext.com/articles/tsiolkovsky-rocket-equation), the [Fenske equation](https://bliptext.com/articles/fenske-equation), or the [Nernst equation](https://bliptext.com/articles/nernst-equation).\n# [A nautilus shell displaying a logarithmic spiral | upload.wikimedia.org/wikipedia/commons/0/08/NautilusCutawayLogarithmicSpiral.jpg]\n\n## Logarithmic scale\nScientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale.\nFor example, the [decibel](https://bliptext.com/articles/decibel) is a [unit of measurement](https://bliptext.com/articles/unit-of-measurement) associated with [logarithmic-scale](https://bliptext.com/articles/logarithmic-scale) [quantities](https://bliptext.com/articles/level-quantity).\nIt is based on the common logarithm of [ratios](https://bliptext.com/articles/ratio)—10 times the common logarithm of a [power](https://bliptext.com/articles/power-physics) ratio or 20 times the common logarithm of a [voltage](https://bliptext.com/articles/voltage) ratio.\nIt is used to quantify the attenuation or amplification of electrical signals, to describe power levels of sounds in [acoustics](https://bliptext.com/articles/acoustics), and the [absorbance](https://bliptext.com/articles/absorbance) of light in the fields of [spectrometry](https://bliptext.com/articles/spectrometer) and [optics](https://bliptext.com/articles/optics).\nThe [[signal](https://bliptext.com/articles/signal-information-theory)-to-[noise](https://bliptext.com/articles/noise-electronic) ratio](https://bliptext.com/articles/[signal](https://bliptext.com/articles/signal-information-theory)-to-[noise](https://bliptext.com/articles/noise-electronic)-ratio) describing the amount of unwanted [noise](https://bliptext.com/articles/noise-electronic) in relation to a (meaningful) [signal](https://bliptext.com/articles/signal-information-theory) is also measured in decibels.\nIn a similar vein, the [peak signal-to-noise ratio](https://bliptext.com/articles/peak-signal-to-noise-ratio) is commonly used to assess the quality of sound and [image compression](https://bliptext.com/articles/image-compression) methods using the logarithm.\nThe strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake.\nThis is used in the [moment magnitude scale](https://bliptext.com/articles/moment-magnitude-scale) or the [Richter magnitude scale](https://bliptext.com/articles/richter-magnitude-scale).\nFor example, a 5.0 earthquake releases 32 times\n(101.5)\nand a 6.0 releases 1000 times\n(103)\nthe energy of a 4.0.\n[Apparent magnitude](https://bliptext.com/articles/apparent-magnitude) measures the brightness of stars logarithmically.\nIn [chemistry](https://bliptext.com/articles/chemistry) the negative of the decimal logarithm, the decimal **cologarithm**, is indicated by the letter p. For instance, [pH](https://bliptext.com/articles/ph) is the decimal **cologarithm** of the [activity](https://bliptext.com/articles/activity-chemistry) of [hydronium](https://bliptext.com/articles/hydronium) [ions](https://bliptext.com/articles/ion) (the form [hydrogen](https://bliptext.com/articles/hydrogen) [ions](https://bliptext.com/articles/ion) take in water).\nThe activity of hydronium ions in neutral water is 10−7 [mol·L−1](https://bliptext.com/articles/molar-concentration), hence a pH of 7.\nVinegar typically has a pH of about 3.\nThe difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about\n10−3 mol·L−1\n.\n[Semilog](https://bliptext.com/articles/semi-log-plot) (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically.\nFor example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million.\nIn such graphs, [exponential functions](https://bliptext.com/articles/exponential-function) of the form\nf(x) = a · b\nappear as straight lines with [slope](https://bliptext.com/articles/slope) equal to the logarithm of b. [Log-log](https://bliptext.com/articles/log-log-plot) graphs scale both axes logarithmically, which causes functions of the form\nf(x) = a · x\nto be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing [power laws](https://bliptext.com/articles/power-law).\n# [A logarithmic chart depicting the value of one Goldmark in Papiermarks during the German hyperinflation in the 1920s | upload.wikimedia.org/wikipedia/commons/4/4f/Germany///Hyperinflation.svg]\n\n## Psychology\nLogarithms occur in several laws describing [human perception](https://bliptext.com/articles/human-perception): [Hick's law](https://bliptext.com/articles/hick-s-law) proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.\n[Fitts's law](https://bliptext.com/articles/fitts-s-law) predicts that the time required to rapidly move to a target area is a logarithmic function of the ratio between the distance to a target and the size of the target.\nIn [psychophysics](https://bliptext.com/articles/psychophysics), the [Weber–Fechner law](https://bliptext.com/articles/weber-fechner-law) proposes a logarithmic relationship between [stimulus](https://bliptext.com/articles/stimulus-psychology) and [sensation](https://bliptext.com/articles/sensation-psychology) such as the actual vs. the perceived weight of an item a person is carrying.\n(This \"law\", however, is less realistic than more recent models, such as [Stevens's power law](https://bliptext.com/articles/stevens-s-power-law). )\nPsychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000.\nIncreasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.\n\n## Probability theory and statistics\nPDF-log normal distributions.svg (PDF) of random variables with log-normal distributions. The location parameter\nμ\n, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.]]\nLogarithms arise in [probability theory](https://bliptext.com/articles/probability-theory): the [law of large numbers](https://bliptext.com/articles/law-of-large-numbers) dictates that, for a [fair coin](https://bliptext.com/articles/fair-coin), as the number of coin-tosses increases to infinity, the observed proportion of heads [approaches one-half](https://bliptext.com/articles/binomial-distribution).\nThe fluctuations of this proportion about one-half are described by the [law of the iterated logarithm](https://bliptext.com/articles/law-of-the-iterated-logarithm).\nLogarithms also occur in [log-normal distributions](https://bliptext.com/articles/log-normal-distribution).\nWhen the logarithm of a [random variable](https://bliptext.com/articles/random-variable) has a [normal distribution](https://bliptext.com/articles/normal-distribution), the variable is said to have a log-[normal distribution](https://bliptext.com/articles/normal-distribution).\nLog-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.\nLogarithms are used for [maximum-likelihood estimation](https://bliptext.com/articles/maximum-likelihood-estimation) of parametric [statistical models](https://bliptext.com/articles/statistical-model).\nFor such a model, the [likelihood function](https://bliptext.com/articles/likelihood-function) depends on at least one [parameter](https://bliptext.com/articles/parametric-model) that must be estimated.\nA maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the \"log likelihood\"), because the logarithm is an increasing function.\nThe log-likelihood is easier to maximize, especially for the multiplied likelihoods for [independent](https://bliptext.com/articles/independence-probability) random variables.\n[Benford's law](https://bliptext.com/articles/benford-s-law) describes the occurrence of digits in many [data sets](https://bliptext.com/articles/data-set), such as heights of buildings.\nAccording to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals\nlog10 (d + 1) − log10 (d)\n, regardless of the unit of measurement.\nThus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.\nThe [logarithm transformation](https://bliptext.com/articles/logarithm-transformation) is a type of [data transformation](https://bliptext.com/articles/data-transformation-statistics) used to bring the empirical distribution closer to the assumed one.\n# [Distribution of first digits (in %, red bars) in the population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law. | upload.wikimedia.org/wikipedia/commons/1/13/Benfords///law///illustrated///by///world%27s///countries///population.svg]\n\n## Computational complexity\n[Analysis of algorithms](https://bliptext.com/articles/analysis-of-[algorithms](https://bliptext.com/articles/algorithm)) is a branch of [computer science](https://bliptext.com/articles/computer-science) that studies the [performance](https://bliptext.com/articles/time-complexity) of [algorithms](https://bliptext.com/articles/algorithm) (computer programs solving a certain problem).\nLogarithms are valuable for describing algorithms that [divide a problem](https://bliptext.com/articles/divide-and-conquer-algorithm) into smaller ones, and join the solutions of the subproblems.\nFor example, to find a number in a sorted list, the [binary search algorithm](https://bliptext.com/articles/binary-search-algorithm) checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found.\nThis algorithm requires, on average,\nlog2 (N)\ncomparisons, where N is the list's length.\nSimilarly, the [merge sort](https://bliptext.com/articles/merge-sort) algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results.\nMerge sort algorithms typically require a time [approximately proportional to](https://bliptext.com/articles/big-o-notation)\nN · log(N)\n. The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used.\nA constant factor is usually disregarded in the analysis of algorithms under the standard [uniform cost model](https://bliptext.com/articles/uniform-cost-model).\nA function\nf(x)\nis said to [grow logarithmically](https://bliptext.com/articles/logarithmic-growth) if\nf(x)\nis (exactly or approximately) proportional to the logarithm of x. (Biological descriptions of organism growth, however, use this term for an exponential function. ) For example, any [natural number](https://bliptext.com/articles/natural-number) N can be represented in [binary form](https://bliptext.com/articles/binary-numeral-system) in no more than\nlog2 N + 1\n[bits](https://bliptext.com/articles/bit).\nIn other words, the amount of [memory](https://bliptext.com/articles/memory-computing) needed to store N grows logarithmically with N.\n\n## Entropy and chaos\n[Entropy](https://bliptext.com/articles/entropy) is broadly a measure of the disorder of some system.\nIn [statistical thermodynamics](https://bliptext.com/articles/statistical-thermodynamics), the entropy S of some physical system is defined as\nThe sum is over all possible states i of the system in question, such as the positions of gas particles in a container.\nMoreover,\npi\nis the probability that the state i is attained and k is the [Boltzmann constant](https://bliptext.com/articles/boltzmann-constant).\nSimilarly, [entropy in information theory](https://bliptext.com/articles/entropy-information-theory) measures the quantity of information.\nIf a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as\nlog2 N\nbits.\n[Lyapunov exponents](https://bliptext.com/articles/lyapunov-exponent) use logarithms to gauge the degree of chaoticity of a [dynamical system](https://bliptext.com/articles/dynamical-system).\nFor example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle.\nSuch systems are [chaotic](https://bliptext.com/articles/chaos-theory) in a [deterministic](https://bliptext.com/articles/deterministic-system) way, because small measurement errors of the initial state predictably lead to largely different final states.\nAt least one Lyapunov exponent of a deterministically chaotic system is positive.\n# [Billiards on an oval billiard table. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of reflections at the boundary. | upload.wikimedia.org/wikipedia/commons/f/fb/Chaotic///Bunimovich///stadium.svg]\n\n## Fractals\nLogarithms occur in definitions of the [dimension](https://bliptext.com/articles/fractal-dimension) of [fractals](https://bliptext.com/articles/fractal).\nFractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure.\nThe [Sierpinski triangle](https://bliptext.com/articles/sierpinski-triangle) (pictured) can be covered by three copies of itself, each having sides half the original length.\nThis makes the [Hausdorff dimension](https://bliptext.com/articles/hausdorff-dimension) of this structure\nln(3)/ln(2) ≈ 1.58\n. Another logarithm-based notion of dimension is obtained by [counting the number of boxes](https://bliptext.com/articles/box-counting-dimension) needed to cover the fractal in question.\n# [400px | upload.wikimedia.org/wikipedia/commons/3/37/Sierpinski///dimension.svg]\n\n## Music\nLogarithms are related to musical tones and [intervals](https://bliptext.com/articles/interval-music).\nIn [equal temperament](https://bliptext.com/articles/equal-temperament) tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or [pitch](https://bliptext.com/articles/pitch-music), of the individual tones.\nIn the [12-tone equal temperament](https://bliptext.com/articles/12-tone-equal-temperament) tuning common in modern Western music, each [octave](https://bliptext.com/articles/octave) (doubling of frequency) is broken into twelve equally spaced intervals called [semitones](https://bliptext.com/articles/semitone).\nFor example, if the note A has a frequency of 440 [Hz](https://bliptext.com/articles/hertz) then the note B-flat has a frequency of 466 [Hz](https://bliptext.com/articles/hertz). The interval between A and B-flat is a [semitone](https://bliptext.com/articles/semitone), as is the one between B-flat and B (frequency 493 [Hz](https://bliptext.com/articles/hertz)).\nAccordingly, the frequency ratios agree:\nIntervals between arbitrary pitches can be measured in octaves by taking the base-\n2 logarithm of the [frequency](https://bliptext.com/articles/frequency) ratio, can be measured in equally tempered semitones by taking the base-\n21/12 logarithm (\n12\ntimes the base-\n2 logarithm), or can be measured in [cents](https://bliptext.com/articles/cent-music), hundredths of a semitone, by taking the base-\n21/1200 logarithm (\n1200\ntimes the base-\n2 logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.\n\n## Number theory\n[Natural logarithms](https://bliptext.com/articles/natural-logarithm) are closely linked to [counting prime numbers](https://bliptext.com/articles/prime-counting-function) (2, 3, 5, 7, 11, ...), an important topic in [number theory](https://bliptext.com/articles/number-theory).\nFor any [integer](https://bliptext.com/articles/integer) x, the quantity of [prime numbers](https://bliptext.com/articles/prime-number) less than or equal to x is denoted\nπ(x)\n. The [prime number theorem](https://bliptext.com/articles/prime-number-theorem) asserts that\nπ(x)\nis approximately given by\nin the sense that the ratio of\nπ(x)\nand that fraction approaches 1 when x tends to infinity.\nAs a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely [proportional](https://bliptext.com/articles/proportionality-mathematics) to the number of decimal digits of x. A far better estimate of\nπ(x)\nis given by the [offset logarithmic integral](https://bliptext.com/articles/logarithmic-integral-function) function\nLi(x)\n, defined by\nThe [Riemann hypothesis](https://bliptext.com/articles/riemann-hypothesis), one of the oldest open mathematical [conjectures](https://bliptext.com/articles/conjecture), can be stated in terms of comparing\nπ(x)\nand\nLi(x)\n. The [Erdős–Kac theorem](https://bliptext.com/articles/erd-s-kac-theorem) describing the number of distinct [prime factors](https://bliptext.com/articles/prime-factor) also involves the [natural logarithm](https://bliptext.com/articles/natural-logarithm).\nThe logarithm of n [factorial](https://bliptext.com/articles/factorial),\nn! = 1 · 2 · ...\n· n\n, is given by\nThis can be used to obtain [Stirling's formula](https://bliptext.com/articles/stirling-s-formula), an approximation of\nn!\nfor large n.\n\n# Generalizations\n\n## Complex logarithm\n[[File:Complex number illustration multiple arguments.svg|thumb|right|Polar form of\n. Both φ and φ' are arguments of z.|alt=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis.]]\nAll the [complex numbers](https://bliptext.com/articles/complex-number) a that solve the equation\ne^a=z\nare called complex logarithms of z, when z is (considered as) a complex number.\nA complex number is commonly represented as\n, where x and y are real numbers and i is an [imaginary unit](https://bliptext.com/articles/imaginary-unit), the square of which is −1.\nSuch a number can be visualized by a point in the [complex plane](https://bliptext.com/articles/complex-plane), as shown at the right.\nThe [polar form](https://bliptext.com/articles/polar-form) encodes a non-zero complex number z by its [absolute value](https://bliptext.com/articles/absolute-value), that is, the (positive, real) distance r to the [origin](https://bliptext.com/articles/origin-mathematics), and an angle between the real (x) axis \nRe\nand the line passing through both the origin and z. This angle is called the [argument](https://bliptext.com/articles/argument-complex-analysis) of z.\nThe absolute value r of z is given by\nUsing the geometrical interpretation of [sine](https://bliptext.com/articles/sine) and co[sine](https://bliptext.com/articles/sine) and their periodicity in\n2π\n, any complex number z may be denoted as\nfor any integer number k. Evidently the argument of z is not uniquely specified: both φ and\nφ' = φ + 2kπ\nare valid arguments of z for all integers k, because adding\n2kπ\n[radians](https://bliptext.com/articles/radian) or k⋅360° to φ corresponds to \"winding\" around the origin counter-clock-wise by k [turns](https://bliptext.com/articles/turn-geometry).\nThe resulting complex number is always z, as illustrated at the right for\n. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted\nArg(z)\n, with a capital\nA\n, by requiring φ to belong to one, conveniently selected turn, e.g.\n−π < φ ≤ π\nor\n0 ≤ φ < 2π\n. These regions, where the argument of z is uniquely determined are called branches of the argument function.\n[[File:Complex log domain.svg|right|thumb|The principal branch (-π, π) of the complex logarithm,\nLog(z)\n. The black point at\ncorresponds to absolute value zero and brighter colors refer to bigger absolute values.\nThe [hue](https://bliptext.com/articles/hue) of the color encodes the argument of\nLog(z)\n.|alt=A density plot.\nIn the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.]]\n[Euler's formula](https://bliptext.com/articles/euler-s-formula) connects the [trigonometric functions](https://bliptext.com/articles/trigonometric-functions) [sine](https://bliptext.com/articles/sine) and co[sine](https://bliptext.com/articles/sine) to the [complex exponential](https://bliptext.com/articles/complex-exponential):\nUsing this formula, and again the periodicity, the following identities hold:\nwhere\nln(r)\nis the unique real natural logarithm,\nak\ndenote the complex logarithms of z, and k is an arbitrary integer.\nTherefore, the complex logarithms of z, which are all those complex values\nak\nfor which the\nak-th\npower of e equals z, are the infinitely many values\nfor arbitrary integers k.\nTaking k such that\nφ + 2kπ\nis within the defined interval for the principal arguments, then\nak\nis called the principal value of the logarithm, denoted\nLog(z)\n, again with a capital\nL\n. The principal argument of any positive real number x is 0; hence\nLog(x)\nis a real number and equals the real (natural) logarithm.\nHowever, the above formulas for logarithms of products and powers do generalize to the principal value of the complex logarithm.\nThe illustration at the right depicts\nLog(z)\n, confining the arguments of z to the interval.\nThis way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there.\nThis discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding k-value of the continuously neighboring branch.\nSuch a locus is called a [branch cut](https://bliptext.com/articles/branch-cut).\nDropping the range restrictions on the argument makes the relations \"argument of z\", and consequently the \"logarithm of z\", [multi-valued functions](https://bliptext.com/articles/multi-valued-function).\n\n## Inverses of other exponential functions\nExponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm.\nFor example, the [logarithm of a matrix](https://bliptext.com/articles/logarithm-of-a-matrix) is the (multi-valued) inverse function of the [matrix exponential](https://bliptext.com/articles/matrix-exponential).\nAnother example is the p-adic logarithm, the inverse function of the p-adic exponential.\nBoth are defined via Taylor series analogous to the real case.\nIn the context of [differential geometry](https://bliptext.com/articles/differential-geometry), the [exponential map](https://bliptext.com/articles/exponential-map-riemannian-geometry) maps the [tangent space](https://bliptext.com/articles/tangent-space) at a point of a [manifold](https://bliptext.com/articles/differentiable-manifold) to a [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) of that point.\nIts inverse is also called the logarithmic (or log) map.\nIn the context of [finite groups](https://bliptext.com/articles/finite-group) exponentiation is given by repeatedly multiplying one group element b with itself.\nThe [discrete logarithm](https://bliptext.com/articles/discrete-logarithm) is the integer n solving the equation\nb^n = x, where x is an element of the group.\nCarrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups.\nThis asymmetry has important applications in [public key cryptography](https://bliptext.com/articles/public-key-cryptography), such as for example in the [Diffie–Hellman key exchange](https://bliptext.com/articles/diffie-hellman-key-exchange), a routine that allows secure exchanges of [cryptographic](https://bliptext.com/articles/cryptography) keys over unsecured information channels.\n[Zech's logarithm](https://bliptext.com/articles/zech-s-logarithm) is related to the discrete logarithm in the multiplicative group of non-zero elements of a [finite field](https://bliptext.com/articles/finite-field).\nFurther logarithm-like inverse functions include the double logarithm\nln(ln(x))\n, the [super- or hyper-4-logarithm](https://bliptext.com/articles/super-logarithm) (a slight variation of which is called [iterated logarithm](https://bliptext.com/articles/iterated-logarithm) in computer science), the [Lambert W function](https://bliptext.com/articles/lambert-w-function), and the [logit](https://bliptext.com/articles/logit).\nThey are the inverse functions of the [double exponential function](https://bliptext.com/articles/double-exponential-function), [tetration](https://bliptext.com/articles/tetration), of\n, and of the [logistic function](https://bliptext.com/articles/logistic-function), respectively.\n\n## Related concepts\nFrom the perspective of [group theory](https://bliptext.com/articles/group-theory), the identity\nexpresses a [group isomorphism](https://bliptext.com/articles/group-isomorphism) between positive [reals](https://bliptext.com/articles/real-number) under multiplication and [reals](https://bliptext.com/articles/real-number) under addition.\nLogarithmic functions are the only continuous isomorphisms between these groups.\nBy means of that isomorphism, the [Haar measure](https://bliptext.com/articles/haar-measure) ([Lebesgue measure](https://bliptext.com/articles/lebesgue-measure))\ndx\non the reals corresponds to the Haar measure\ndx/x\non the positive reals.\nThe non-negative reals not only have a multiplication, but also have addition, and form a [semiring](https://bliptext.com/articles/semiring), called the probability [semiring](https://bliptext.com/articles/semiring); this is in fact a [semifield](https://bliptext.com/articles/semifield).\nThe logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition ([LogSumExp](https://bliptext.com/articles/logsumexp)), giving an [isomorphism](https://bliptext.com/articles/isomorphism) of semirings between the probability semiring and the [log semiring](https://bliptext.com/articles/log-semiring).\nLogarithmic one-forms\ndf/f\nappear in [complex analysis](https://bliptext.com/articles/complex-analysis) and [algebraic geometry](https://bliptext.com/articles/algebraic-geometry) as [differential forms](https://bliptext.com/articles/differential-form) with logarithmic [poles](https://bliptext.com/articles/pole-complex-analysis).\nThe [polylogarithm](https://bliptext.com/articles/polylogarithm) is the function defined by\nIt is related to the [natural logarithm](https://bliptext.com/articles/natural-logarithm) by\nLi1 (z) = −ln(1 − z)\n. Moreover,\nLis (1)\nequals the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function)\nζ(s)\n.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:43:04.592Z","updated_at":"2025-01-25T16:43:04.592Z","search_vector":"'logarithm':1","search_content":" \n \n\n# \n[[File:Logarithm plots.png|right|thumb|upright=1.35|Plots of logarithm functions, with three commonly used bases. The special points\nlogb b = 1\nare indicated by dotted lines, and all curves intersect in\nlogb 1 = 0\n.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the **logarithm** to ****base**** b is the inverse function of [[exponent](https://bliptext.com/articles/exponent)iation](https://bliptext.com/articles/[exponent](https://bliptext.com/articles/exponent)iation) with ****base**** b. That means that the **logarithm** of a number x to the ****base**** b is the [exponent](https://bliptext.com/articles/exponent) to which b must be raised to produce x. For example, since\n, the logarithm base 10 of\n1000\nis\n3\n, or\nlog10 (1000) = 3\n. The logarithm of x to base b is denoted as\nlogb (x)\n, or without parentheses,\nlogb x\n. When the base is clear from the context or is irrelevant it is sometimes written\nlog x\n.\nThe logarithm base\n10\nis called the decimal or common logarithm and is commonly used in science and engineering.\nThe natural logarithm has the number [[e (mathematical constant)|\ne ≈ 2.718\n]] as its base; its use is widespread in mathematics and [physics](https://bliptext.com/articles/physics) because of its very simple [derivative](https://bliptext.com/articles/derivative).\nThe binary logarithm uses base\n2\nand is frequently used in [computer science](https://bliptext.com/articles/computer-science).\nLogarithms were introduced by [John Napier](https://bliptext.com/articles/john-napier) in 1614 as a means of simplifying calculations.\nThey were rapidly adopted by [navigators](https://bliptext.com/articles/navigator), scientists, engineers, [surveyors](https://bliptext.com/articles/surveying), and others to perform high-accuracy computations more easily.\nUsing [logarithm tables](https://bliptext.com/articles/logarithm-table), tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition.\nThis is possible because the logarithm of a [product](https://bliptext.com/articles/product-mathematics) is the [sum](https://bliptext.com/articles/summation) of the logarithms of the factors:\nprovided that b, x and y are all positive and\nb ≠ 1\n. The [slide rule](https://bliptext.com/articles/slide-rule), also based on logarithms, allows quick calculations without tables, but at lower precision.\nThe present-day notion of logarithms comes from [Leonhard Euler](https://bliptext.com/articles/leonhard-euler), who connected them to the [exponential function](https://bliptext.com/articles/exponential-function) in the 18th century, and who also introduced the letter e as the base of natural logarithms.\n[Logarithmic scales](https://bliptext.com/articles/logarithmic-scale) reduce wide-ranging quantities to smaller scopes.\nFor example, the [decibel](https://bliptext.com/articles/decibel) (dB) is a [unit](https://bliptext.com/articles/units-of-measurement) used to express [ratio as logarithms](https://bliptext.com/articles/level-logarithmic-quantity), mostly for signal power and amplitude (of which [sound pressure](https://bliptext.com/articles/sound-pressure) is a common example).\nIn chemistry, [pH](https://bliptext.com/articles/ph) is a logarithmic measure for the [acidity](https://bliptext.com/articles/acid) of an [aqueous solution](https://bliptext.com/articles/aqueous-solution).\nLogarithms are commonplace in scientific [formulae](https://bliptext.com/articles/formula), and in measurements of the [complexity of algorithms](https://bliptext.com/articles/computational-complexity-theory) and of geometric objects called [fractals](https://bliptext.com/articles/fractal).\nThey help to describe [frequency](https://bliptext.com/articles/frequency) ratios of [musical intervals](https://bliptext.com/articles/interval-music), appear in formulas counting [prime numbers](https://bliptext.com/articles/prime-number) or [approximating](https://bliptext.com/articles/stirling-s-approximation) [factorials](https://bliptext.com/articles/factorial), inform some models in [psychophysics](https://bliptext.com/articles/psychophysics), and can aid in [forensic accounting](https://bliptext.com/articles/forensic-accounting).\nThe concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well.\nHowever, in general settings, the logarithm tends to be a multi-valued function.\nFor example, the [complex logarithm](https://bliptext.com/articles/complex-logarithm) is the multi-valued [inverse](https://bliptext.com/articles/inverse-function) of the complex exponential function.\nSimilarly, the [discrete logarithm](https://bliptext.com/articles/discrete-logarithm) is the multi-valued inverse of the exponential function in finite groups; it has uses in [public-key cryptography](https://bliptext.com/articles/public-key-cryptography).\n\n# Motivation\nBinary logarithm plot with grid.png of the logarithm base 2 crosses the x-axis at\nand passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g.,\nand\n. The graph gets arbitrarily close to the y-axis, but [does not meet it](https://bliptext.com/articles/asymptotic).]]\n[Addition](https://bliptext.com/articles/addition), [multiplication](https://bliptext.com/articles/multiplication), and [exponentiation](https://bliptext.com/articles/exponentiation) are three of the most fundamental arithmetic operations.\nThe inverse of addition is [subtraction](https://bliptext.com/articles/subtraction), and the inverse of multiplication is [division](https://bliptext.com/articles/division-mathematics).\nSimilarly, a logarithm is the inverse operation of [exponentiation](https://bliptext.com/articles/exponentiation).\nExponentiation is when a number b, the base, is raised to a certain power y, the exponent, to give a value x; this is denoted\nb^y=x.\nFor example, raising\n2\nto the power of\n3\ngives\n8\nThe logarithm of base b is the inverse operation, that provides the output y from the input x. That is, is equivalent to x=b^y if b is a positive [real number](https://bliptext.com/articles/real-number).\n(If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)\nOne of the main historical motivations of introducing logarithms is the formula\nby which [tables of logarithms](https://bliptext.com/articles/logarithm-table) allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.\n\n# Definition\nGiven a positive [real number](https://bliptext.com/articles/real-number) b such that\nb ≠ 1\n, the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that b^y = x.\nThe logarithm is denoted \"\nlogb x\n\" (pronounced as \"the logarithm of x to base b\", \"the base-b logarithm of x\", or most commonly \"the log, base b, of x\").\nAn equivalent and more succinct definition is that the function\nlogb\nis the [inverse function](https://bliptext.com/articles/inverse-function) to the function.\n\n## Examples\nlog2 16 = 4\n, since\n24 = 2 × 2 × 2 × 2 = 16\n.\nlog10 150\nis approximately 2.176, which lies between 2 and 3, just as 150 lies between\n102 = 100\nand\n103 = 1000\n.\nlogb b = 1\nand\nlogb 1 = 0\n, since\nb1 = b\nand\nb0 = 1\n, respectively.\n\n# Logarithmic identities\nSeveral important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.\n\n## Product, quotient, power, and root\nThe logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms.\nThe logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples.\nEach of the identities can be derived after substitution of the logarithm definitions or in the left hand sides.\n\n## Change of base\nThe logarithm\nlogb x\ncan be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:\nTypical [scientific calculators](https://bliptext.com/articles/scientific-calculators) calculate the logarithms to bases 10 and e. Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:\nGiven a number x and its logarithm\ny = logb x\nto an unknown base b, the base is given by:\nwhich can be seen from taking the defining equation to the power of\n\n# Particular bases \nAmong all choices for the base, three are particularly common.\nThese are\n,\n(the [irrational](https://bliptext.com/articles/irrational-number) mathematical constant e ≈ 2.71828183\n), and\n(the [binary logarithm](https://bliptext.com/articles/binary-logarithm)).\nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), the logarithm base e is widespread because of analytical properties explained below.\nOn the other hand, base 10 logarithms (the [common logarithm](https://bliptext.com/articles/common-logarithm)) are easy to use for manual calculations in the [decimal](https://bliptext.com/articles/decimal) number system:\nThus,\nlog10 (x)\nis related to the number of [decimal digits](https://bliptext.com/articles/decimal-digit) of a positive [integer](https://bliptext.com/articles/integer) x: The number of digits is the smallest [integer](https://bliptext.com/articles/integer) strictly bigger than log10 (x)\n. For example,\nlog10(5986)\nis approximately 3.78.\nThe [next integer above](https://bliptext.com/articles/floor-and-ceiling-functions) it is 4, which is the number of digits of 5986.\nBoth the natural logarithm and the binary logarithm are used in [information theory](https://bliptext.com/articles/information-theory), corresponding to the use of [nats](https://bliptext.com/articles/nat-unit) or [bits](https://bliptext.com/articles/bit) as the fundamental units of information, respectively.\nBinary logarithms are also used in [computer science](https://bliptext.com/articles/computer-science), where the [binary system](https://bliptext.com/articles/binary-numeral-system) is ubiquitous; in [music theory](https://bliptext.com/articles/music-theory), where a pitch ratio of two (the [octave](https://bliptext.com/articles/octave)) is ubiquitous and the number of [cents](https://bliptext.com/articles/cent-music) between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 [cents](https://bliptext.com/articles/cent-music) per [semitone](https://bliptext.com/articles/semitone) in [conventional equal temperament](https://bliptext.com/articles/12-tone-equal-temperament)), or equivalently the log base 21/1200\n; and in [photography](https://bliptext.com/articles/photography) rescaled base 2 logarithms are used to measure [exposure values](https://bliptext.com/articles/exposure-value), [light levels](https://bliptext.com/articles/luminance), [exposure times](https://bliptext.com/articles/exposure-time), lens [apertures](https://bliptext.com/articles/aperture), and [film speeds](https://bliptext.com/articles/film-speed) in \"stops\".\nThe abbreviation\nlog x\nis often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial.\nCommon logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts\nlog x\nstill often means the base ten logarithm.\nIn mathematics\nlog x\nusually means to the natural logarithm (base e).\nIn computer science and information theory,\nlog\noften refers to binary logarithms (base 2).\nThe following table lists common notations for logarithms to these bases.\nThe \"ISO notation\" column lists designations suggested by the [International Organization for Standardization](https://bliptext.com/articles/international-organization-for-standardization).\n# [Overlaid graphs of the logarithm for bases 1⁄2, 2, and e | upload.wikimedia.org/wikipedia/commons/a/ae/Log4.svg]\n\n# History\nThe history of logarithms in seventeenth-century Europe saw the discovery of a new [function](https://bliptext.com/articles/function-mathematics) that extended the realm of analysis beyond the scope of algebraic methods.\nThe method of logarithms was publicly propounded by [John Napier](https://bliptext.com/articles/john-napier) in 1614, in a book titled [Mirifici Logarithmorum Canonis Descriptio](https://bliptext.com/articles/mirifici-logarithmorum-canonis-descriptio) (Description of the Wonderful Canon of Logarithms).\nPrior to Napier's invention, there had been other techniques of similar scopes, such as the [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis) or the use of tables of progressions, extensively developed by [Jost Bürgi](https://bliptext.com/articles/jost-b-rgi) around 1600.\nNapier coined the term for logarithm in Middle Latin, logarithmus, literally meaning ratio-number, derived from the Greek logos proportion, ratio, word + arithmos number.\nThe [common logarithm](https://bliptext.com/articles/common-logarithm) of a number is the index of that power of ten which equals the number.\nSpeaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by [Archimedes](https://bliptext.com/articles/archimedes) as the \"order of a number\".\nThe first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation.\nSome of these methods used tables derived from trigonometric identities.\nSuch methods are called [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis).\nInvention of the [function](https://bliptext.com/articles/function-mathematics) now known as the [natural logarithm](https://bliptext.com/articles/natural-logarithm) began as an attempt to perform a [quadrature](https://bliptext.com/articles/quadrature-mathematics) of a rectangular [hyperbola](https://bliptext.com/articles/hyperbola) by [Grégoire de Saint-Vincent](https://bliptext.com/articles/gr-goire-de-saint-vincent), a Belgian Jesuit residing in Prague.\nArchimedes had written [The Quadrature of the Parabola](https://bliptext.com/articles/the-quadrature-of-the-parabola) in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647.\nThe relation that the logarithm provides between a [geometric progression](https://bliptext.com/articles/geometric-progression) in its [argument](https://bliptext.com/articles/argument-of-a-function) and an [arithmetic progression](https://bliptext.com/articles/arithmetic-progression) of values, prompted [A. A. de Sarasa](https://bliptext.com/articles/a-a-de-sarasa) to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis), leading to the term \"hyperbolic logarithm\", a synonym for natural logarithm.\nSoon the new function was appreciated by [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens), and [James Gregory](https://bliptext.com/articles/james-gregory-mathematician).\nThe notation\nLog y\nwas adopted by [Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz) in 1675, and the next year he connected it to the [integral](https://bliptext.com/articles/integral-calculus)\nBefore Euler developed his modern conception of complex natural logarithms, [Roger Cotes](https://bliptext.com/articles/roger-cotes) had a nearly equivalent result when he showed in 1714 that\n\n# Logarithm tables, slide rules, and historical applications\nBy simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially [astronomy](https://bliptext.com/articles/astronomy).\nThey were critical to advances in [surveying](https://bliptext.com/articles/surveying), [celestial navigation](https://bliptext.com/articles/celestial-navigation), and other domains.\n[Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace) called logarithms\nAs the function\nis the inverse function of\nlogb x\n, it has been called an **antilogarithm**.\nNowadays, this function is more commonly called an [exponential function](https://bliptext.com/articles/exponential-function).\n# [The 1797 Encyclopædia Britannica explanation of logarithms | upload.wikimedia.org/wikipedia/commons/8/82/Logarithms///Britannica///1797.png]\n\n## Log tables\nA key tool that enabled the practical use of logarithms was the [table of logarithms](https://bliptext.com/articles/log-table).\nThe first such table was compiled by [Henry Briggs](https://bliptext.com/articles/henry-briggs-mathematician) in 1617, immediately after Napier's invention but with the innovation of using 10 as the base.\nBriggs' first table contained the [common logarithms](https://bliptext.com/articles/common-logarithm) of all integers in the range from 1 to 1000, with a precision of 14 digits.\nSubsequently, tables with increasing scope were written.\nThese tables listed the values of\nlog10 x\nfor any number x in a certain range, at a certain precision.\nBase-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers.\nThe common logarithm of x can be separated into an [integer part](https://bliptext.com/articles/integer-part) and a [fractional part](https://bliptext.com/articles/fractional-part), known as the characteristic and [mantissa](https://bliptext.com/articles/mantissa-logarithm).\nTables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.\nThe characteristic of\n10 · x\nis one plus the characteristic of x, and their mantissas are the same.\nThus using a three-digit log table, the logarithm of 3542 is approximated by\nGreater accuracy can be obtained by [interpolation](https://bliptext.com/articles/interpolation):\nThe value of\n10x\ncan be determined by reverse look up in the same table, since the logarithm is a [monotonic function](https://bliptext.com/articles/monotonic-function).\n\n## Computations\nThe product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms.\nThe product\ncd\nor quotient\nc/d\ncame from looking up the antilogarithm of the sum or difference, via the same table:\nand\nFor manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as [prosthaphaeresis](https://bliptext.com/articles/prosthaphaeresis), which relies on [trigonometric identities](https://bliptext.com/articles/trigonometric-identities).\nCalculations of powers and [roots](https://bliptext.com/articles/nth-root) are reduced to multiplications or divisions and lookups by\nand\nTrigonometric calculations were facilitated by tables that contained the common logarithms of [trigonometric functions](https://bliptext.com/articles/trigonometric-function).\n\n## Slide rules\nAnother critical application was the [slide rule](https://bliptext.com/articles/slide-rule), a pair of logarithmically divided scales used for calculation.\nThe non-sliding logarithmic scale, [Gunter's rule](https://bliptext.com/articles/gunter-s-rule), was invented shortly after Napier's invention.\n[William Oughtred](https://bliptext.com/articles/william-oughtred) enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other.\nNumbers are placed on sliding scales at distances proportional to the differences between their logarithms.\nSliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:\nFor example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part.\nThe slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.\n# [Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x. | upload.wikimedia.org/wikipedia/commons/8/8f/Slide///rule///example2///with///labels.svg]\n\n# Analytic properties\nA deeper study of logarithms requires the concept of a [function](https://bliptext.com/articles/function-mathematics).\nA function is a rule that, given one number, produces another number.\nAn example is the function producing the x-th power of b from any real number x, where the base b is a fixed number.\nThis function is written as\nf(x) = b x\n. When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals.\n\n## Existence\nLet b be a positive real number not equal to 1 and let\nf(x) = b x\n.\nIt is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range.\nThis fact follows from the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem).\nNow, f is [strictly increasing](https://bliptext.com/articles/monotonic-function) (for\nb > 1\n), or strictly decreasing (for\n0 0}. Therefore, f is a bijection from \\R to \\R_{>0}. In other words, for each positive real number y, there is exactly one real number x such that b^x = y.\nWe let denote the inverse of f. That is,\nlogb y\nis the unique real number x such that b^x = y. This function is called the base-b logarithm function or logarithmic function (or just logarithm).\n\n## Characterization by the product formula\nThe function\nlogb x\ncan also be essentially characterized by the product formula\nMore precisely, the logarithm to any base\nb > 1\nis the only [increasing function](https://bliptext.com/articles/increasing-function) f from the positive reals to the reals satisfying\nf(b) = 1\nand\n\n## Graph of the logarithm function\n[[File:Logarithm inversefunctiontoexp.svg|right|thumb|The graph of the logarithm function\nlogb (x)\n(blue) is obtained by [reflecting](https://bliptext.com/articles/reflection-mathematics) the graph of the function\nbx\n(red) at the diagonal line (\nx = y\n).|alt=The graphs of two functions.]] As discussed above, the function\nlogb\nis the inverse to the exponential function.\nTherefore, their [graphs](https://bliptext.com/articles/graph-of-a-function) correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line\nx = y\n), as shown at the right: a point\n(t, u = bt)\non the graph of f yields a point\n(u, t = logb u)\non the graph of the logarithm and vice versa.\nAs a consequence,\nlogb (x)\n[diverges to infinity](https://bliptext.com/articles/divergent-sequence) (gets bigger than any given number) if x grows to infinity, provided that b is greater than one.\nIn that case,\nlogb(x)\nis an [increasing function](https://bliptext.com/articles/increasing-function).\nFor\nb 1\n(plus infinity for\nb 0\n. Using [sigma notation](https://bliptext.com/articles/sigma-notation), this is also written as\nThis series can be derived from the above Taylor series.\nIt converges quicker than the Taylor series, especially if z is close to 1.\nFor example, for\nz = 1.5\n, the first three terms of the second series approximate\nln(1.5)\nwith an error of about 3.\nThe quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation\ny ≈ ln(z)\nand putting\nthe logarithm of z is:\nThe better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently.\nA can be calculated using the [exponential series](https://bliptext.com/articles/exponential-function), which converges quickly provided y is not too large.\nCalculating the logarithm of larger z can be reduced to smaller values of z by writing\n, so that\n.\nA closely related method can be used to compute the logarithm of integers.\nPutting in the above series, it follows that:\nIf the logarithm of a large integer n is known, then this series yields a fast converging series for\nlog(n+1)\n, with a [rate of convergence](https://bliptext.com/articles/rate-of-convergence) of.\n\n## Arithmetic–geometric mean approximation\nThe [arithmetic–geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean) yields high-precision approximations of the [natural logarithm](https://bliptext.com/articles/natural-logarithm).\nSasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed.\nIn their work\nln(x)\nis approximated to a precision of\n2−p\n(or p precise bits) by the following formula (due to [Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss)):\nHere\nM(x, y)\ndenotes the [arithmetic–geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean) of x and y. It is obtained by repeatedly calculating the average\n(x + y)/2\n([arithmetic mean](https://bliptext.com/articles/arithmetic-mean)) and \\sqrt{xy} ([geometric mean](https://bliptext.com/articles/geometric-mean)) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of\nM(x, y)\n. m is chosen such that\nto ensure the required precision.\nA larger m makes the\nM(x, y)\ncalculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision.\nThe constants\nπ\nand\nln(2)\ncan be calculated with quickly converging series.\n\n## Feynman's algorithm\nWhile at [Los Alamos National Laboratory](https://bliptext.com/articles/los-alamos-national-laboratory) working on the [Manhattan Project](https://bliptext.com/articles/manhattan-project), [Richard Feynman](https://bliptext.com/articles/richard-feynman) developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in the [Connection Machine](https://bliptext.com/articles/connection-machine).\nThe algorithm relies on the fact that every real number x where\n1 < x < 2\ncan be represented as a product of distinct factors of the form\n1 + 2−k\n. The algorithm sequentially builds that product P, starting with\nand\nP · (1 + 2−k) < x\n, then it changes P to\nP · (1 + 2−k)\n. It then increases k by one regardless.\nThe algorithm stops when k is large enough to give the desired accuracy.\nBecause\nlog(x)\nis the sum of the terms of the form\nlog(1 + 2−k)\ncorresponding to those k for which the factor\n1 + 2−k\nwas included in the product P,\nlog(x)\nmay be computed by simple addition, using a table of\nlog(1 + 2−k)\nfor all k. Any base may be used for the logarithm table.\n\n# Applications\nLogarithms have many applications inside and outside mathematics.\nSome of these occurrences are related to the notion of [scale invariance](https://bliptext.com/articles/scale-invariance).\nFor example, each chamber of the shell of a [nautilus](https://bliptext.com/articles/nautilus) is an approximate copy of the next one, scaled by a constant factor.\nThis gives rise to a [logarithmic spiral](https://bliptext.com/articles/logarithmic-spiral).\n[Benford's law](https://bliptext.com/articles/benford-s-law) on the distribution of leading digits can also be explained by scale invariance.\nLogarithms are also linked to [self-similarity](https://bliptext.com/articles/self-similarity).\nFor example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.\nThe dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.\n[Logarithmic scales](https://bliptext.com/articles/logarithmic-scale) are useful for quantifying the relative change of a value as opposed to its absolute difference.\nMoreover, because the logarithmic function\nlog(x)\ngrows very slowly for large x, logarithmic scales are used to compress large-scale scientific data.\nLogarithms also occur in numerous scientific formulas, such as the [Tsiolkovsky rocket equation](https://bliptext.com/articles/tsiolkovsky-rocket-equation), the [Fenske equation](https://bliptext.com/articles/fenske-equation), or the [Nernst equation](https://bliptext.com/articles/nernst-equation).\n# [A nautilus shell displaying a logarithmic spiral | upload.wikimedia.org/wikipedia/commons/0/08/NautilusCutawayLogarithmicSpiral.jpg]\n\n## Logarithmic scale\nScientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale.\nFor example, the [decibel](https://bliptext.com/articles/decibel) is a [unit of measurement](https://bliptext.com/articles/unit-of-measurement) associated with [logarithmic-scale](https://bliptext.com/articles/logarithmic-scale) [quantities](https://bliptext.com/articles/level-quantity).\nIt is based on the common logarithm of [ratios](https://bliptext.com/articles/ratio)—10 times the common logarithm of a [power](https://bliptext.com/articles/power-physics) ratio or 20 times the common logarithm of a [voltage](https://bliptext.com/articles/voltage) ratio.\nIt is used to quantify the attenuation or amplification of electrical signals, to describe power levels of sounds in [acoustics](https://bliptext.com/articles/acoustics), and the [absorbance](https://bliptext.com/articles/absorbance) of light in the fields of [spectrometry](https://bliptext.com/articles/spectrometer) and [optics](https://bliptext.com/articles/optics).\nThe [[signal](https://bliptext.com/articles/signal-information-theory)-to-[noise](https://bliptext.com/articles/noise-electronic) ratio](https://bliptext.com/articles/[signal](https://bliptext.com/articles/signal-information-theory)-to-[noise](https://bliptext.com/articles/noise-electronic)-ratio) describing the amount of unwanted [noise](https://bliptext.com/articles/noise-electronic) in relation to a (meaningful) [signal](https://bliptext.com/articles/signal-information-theory) is also measured in decibels.\nIn a similar vein, the [peak signal-to-noise ratio](https://bliptext.com/articles/peak-signal-to-noise-ratio) is commonly used to assess the quality of sound and [image compression](https://bliptext.com/articles/image-compression) methods using the logarithm.\nThe strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake.\nThis is used in the [moment magnitude scale](https://bliptext.com/articles/moment-magnitude-scale) or the [Richter magnitude scale](https://bliptext.com/articles/richter-magnitude-scale).\nFor example, a 5.0 earthquake releases 32 times\n(101.5)\nand a 6.0 releases 1000 times\n(103)\nthe energy of a 4.0.\n[Apparent magnitude](https://bliptext.com/articles/apparent-magnitude) measures the brightness of stars logarithmically.\nIn [chemistry](https://bliptext.com/articles/chemistry) the negative of the decimal logarithm, the decimal **cologarithm**, is indicated by the letter p. For instance, [pH](https://bliptext.com/articles/ph) is the decimal **cologarithm** of the [activity](https://bliptext.com/articles/activity-chemistry) of [hydronium](https://bliptext.com/articles/hydronium) [ions](https://bliptext.com/articles/ion) (the form [hydrogen](https://bliptext.com/articles/hydrogen) [ions](https://bliptext.com/articles/ion) take in water).\nThe activity of hydronium ions in neutral water is 10−7 [mol·L−1](https://bliptext.com/articles/molar-concentration), hence a pH of 7.\nVinegar typically has a pH of about 3.\nThe difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about\n10−3 mol·L−1\n.\n[Semilog](https://bliptext.com/articles/semi-log-plot) (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically.\nFor example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million.\nIn such graphs, [exponential functions](https://bliptext.com/articles/exponential-function) of the form\nf(x) = a · b\nappear as straight lines with [slope](https://bliptext.com/articles/slope) equal to the logarithm of b. [Log-log](https://bliptext.com/articles/log-log-plot) graphs scale both axes logarithmically, which causes functions of the form\nf(x) = a · x\nto be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing [power laws](https://bliptext.com/articles/power-law).\n# [A logarithmic chart depicting the value of one Goldmark in Papiermarks during the German hyperinflation in the 1920s | upload.wikimedia.org/wikipedia/commons/4/4f/Germany///Hyperinflation.svg]\n\n## Psychology\nLogarithms occur in several laws describing [human perception](https://bliptext.com/articles/human-perception): [Hick's law](https://bliptext.com/articles/hick-s-law) proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.\n[Fitts's law](https://bliptext.com/articles/fitts-s-law) predicts that the time required to rapidly move to a target area is a logarithmic function of the ratio between the distance to a target and the size of the target.\nIn [psychophysics](https://bliptext.com/articles/psychophysics), the [Weber–Fechner law](https://bliptext.com/articles/weber-fechner-law) proposes a logarithmic relationship between [stimulus](https://bliptext.com/articles/stimulus-psychology) and [sensation](https://bliptext.com/articles/sensation-psychology) such as the actual vs. the perceived weight of an item a person is carrying.\n(This \"law\", however, is less realistic than more recent models, such as [Stevens's power law](https://bliptext.com/articles/stevens-s-power-law). )\nPsychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000.\nIncreasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.\n\n## Probability theory and statistics\nPDF-log normal distributions.svg (PDF) of random variables with log-normal distributions. The location parameter\nμ\n, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.]]\nLogarithms arise in [probability theory](https://bliptext.com/articles/probability-theory): the [law of large numbers](https://bliptext.com/articles/law-of-large-numbers) dictates that, for a [fair coin](https://bliptext.com/articles/fair-coin), as the number of coin-tosses increases to infinity, the observed proportion of heads [approaches one-half](https://bliptext.com/articles/binomial-distribution).\nThe fluctuations of this proportion about one-half are described by the [law of the iterated logarithm](https://bliptext.com/articles/law-of-the-iterated-logarithm).\nLogarithms also occur in [log-normal distributions](https://bliptext.com/articles/log-normal-distribution).\nWhen the logarithm of a [random variable](https://bliptext.com/articles/random-variable) has a [normal distribution](https://bliptext.com/articles/normal-distribution), the variable is said to have a log-[normal distribution](https://bliptext.com/articles/normal-distribution).\nLog-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.\nLogarithms are used for [maximum-likelihood estimation](https://bliptext.com/articles/maximum-likelihood-estimation) of parametric [statistical models](https://bliptext.com/articles/statistical-model).\nFor such a model, the [likelihood function](https://bliptext.com/articles/likelihood-function) depends on at least one [parameter](https://bliptext.com/articles/parametric-model) that must be estimated.\nA maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the \"log likelihood\"), because the logarithm is an increasing function.\nThe log-likelihood is easier to maximize, especially for the multiplied likelihoods for [independent](https://bliptext.com/articles/independence-probability) random variables.\n[Benford's law](https://bliptext.com/articles/benford-s-law) describes the occurrence of digits in many [data sets](https://bliptext.com/articles/data-set), such as heights of buildings.\nAccording to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals\nlog10 (d + 1) − log10 (d)\n, regardless of the unit of measurement.\nThus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.\nThe [logarithm transformation](https://bliptext.com/articles/logarithm-transformation) is a type of [data transformation](https://bliptext.com/articles/data-transformation-statistics) used to bring the empirical distribution closer to the assumed one.\n# [Distribution of first digits (in %, red bars) in the population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law. | upload.wikimedia.org/wikipedia/commons/1/13/Benfords///law///illustrated///by///world%27s///countries///population.svg]\n\n## Computational complexity\n[Analysis of [algorithms](https://bliptext.com/articles/algorithm)](https://bliptext.com/articles/analysis-of-[algorithms](https://bliptext.com/articles/algorithm)) is a branch of [computer science](https://bliptext.com/articles/computer-science) that studies the [performance](https://bliptext.com/articles/time-complexity) of [algorithms](https://bliptext.com/articles/algorithm) (computer programs solving a certain problem).\nLogarithms are valuable for describing algorithms that [divide a problem](https://bliptext.com/articles/divide-and-conquer-algorithm) into smaller ones, and join the solutions of the subproblems.\nFor example, to find a number in a sorted list, the [binary search algorithm](https://bliptext.com/articles/binary-search-algorithm) checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found.\nThis algorithm requires, on average,\nlog2 (N)\ncomparisons, where N is the list's length.\nSimilarly, the [merge sort](https://bliptext.com/articles/merge-sort) algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results.\nMerge sort algorithms typically require a time [approximately proportional to](https://bliptext.com/articles/big-o-notation)\nN · log(N)\n. The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used.\nA constant factor is usually disregarded in the analysis of algorithms under the standard [uniform cost model](https://bliptext.com/articles/uniform-cost-model).\nA function\nf(x)\nis said to [grow logarithmically](https://bliptext.com/articles/logarithmic-growth) if\nf(x)\nis (exactly or approximately) proportional to the logarithm of x. (Biological descriptions of organism growth, however, use this term for an exponential function. ) For example, any [natural number](https://bliptext.com/articles/natural-number) N can be represented in [binary form](https://bliptext.com/articles/binary-numeral-system) in no more than\nlog2 N + 1\n[bits](https://bliptext.com/articles/bit).\nIn other words, the amount of [memory](https://bliptext.com/articles/memory-computing) needed to store N grows logarithmically with N.\n\n## Entropy and chaos\n[Entropy](https://bliptext.com/articles/entropy) is broadly a measure of the disorder of some system.\nIn [statistical thermodynamics](https://bliptext.com/articles/statistical-thermodynamics), the entropy S of some physical system is defined as\nThe sum is over all possible states i of the system in question, such as the positions of gas particles in a container.\nMoreover,\npi\nis the probability that the state i is attained and k is the [Boltzmann constant](https://bliptext.com/articles/boltzmann-constant).\nSimilarly, [entropy in information theory](https://bliptext.com/articles/entropy-information-theory) measures the quantity of information.\nIf a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as\nlog2 N\nbits.\n[Lyapunov exponents](https://bliptext.com/articles/lyapunov-exponent) use logarithms to gauge the degree of chaoticity of a [dynamical system](https://bliptext.com/articles/dynamical-system).\nFor example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle.\nSuch systems are [chaotic](https://bliptext.com/articles/chaos-theory) in a [deterministic](https://bliptext.com/articles/deterministic-system) way, because small measurement errors of the initial state predictably lead to largely different final states.\nAt least one Lyapunov exponent of a deterministically chaotic system is positive.\n# [Billiards on an oval billiard table. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of reflections at the boundary. | upload.wikimedia.org/wikipedia/commons/f/fb/Chaotic///Bunimovich///stadium.svg]\n\n## Fractals\nLogarithms occur in definitions of the [dimension](https://bliptext.com/articles/fractal-dimension) of [fractals](https://bliptext.com/articles/fractal).\nFractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure.\nThe [Sierpinski triangle](https://bliptext.com/articles/sierpinski-triangle) (pictured) can be covered by three copies of itself, each having sides half the original length.\nThis makes the [Hausdorff dimension](https://bliptext.com/articles/hausdorff-dimension) of this structure\nln(3)/ln(2) ≈ 1.58\n. Another logarithm-based notion of dimension is obtained by [counting the number of boxes](https://bliptext.com/articles/box-counting-dimension) needed to cover the fractal in question.\n# [400px | upload.wikimedia.org/wikipedia/commons/3/37/Sierpinski///dimension.svg]\n\n## Music\nLogarithms are related to musical tones and [intervals](https://bliptext.com/articles/interval-music).\nIn [equal temperament](https://bliptext.com/articles/equal-temperament) tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or [pitch](https://bliptext.com/articles/pitch-music), of the individual tones.\nIn the [12-tone equal temperament](https://bliptext.com/articles/12-tone-equal-temperament) tuning common in modern Western music, each [octave](https://bliptext.com/articles/octave) (doubling of frequency) is broken into twelve equally spaced intervals called [semitones](https://bliptext.com/articles/semitone).\nFor example, if the note A has a frequency of 440 [Hz](https://bliptext.com/articles/hertz) then the note B-flat has a frequency of 466 [Hz](https://bliptext.com/articles/hertz). The interval between A and B-flat is a [semitone](https://bliptext.com/articles/semitone), as is the one between B-flat and B (frequency 493 [Hz](https://bliptext.com/articles/hertz)).\nAccordingly, the frequency ratios agree:\nIntervals between arbitrary pitches can be measured in octaves by taking the base-\n2 logarithm of the [frequency](https://bliptext.com/articles/frequency) ratio, can be measured in equally tempered semitones by taking the base-\n21/12 logarithm (\n12\ntimes the base-\n2 logarithm), or can be measured in [cents](https://bliptext.com/articles/cent-music), hundredths of a semitone, by taking the base-\n21/1200 logarithm (\n1200\ntimes the base-\n2 logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.\n\n## Number theory\n[Natural logarithms](https://bliptext.com/articles/natural-logarithm) are closely linked to [counting prime numbers](https://bliptext.com/articles/prime-counting-function) (2, 3, 5, 7, 11, ...), an important topic in [number theory](https://bliptext.com/articles/number-theory).\nFor any [integer](https://bliptext.com/articles/integer) x, the quantity of [prime numbers](https://bliptext.com/articles/prime-number) less than or equal to x is denoted\nπ(x)\n. The [prime number theorem](https://bliptext.com/articles/prime-number-theorem) asserts that\nπ(x)\nis approximately given by\nin the sense that the ratio of\nπ(x)\nand that fraction approaches 1 when x tends to infinity.\nAs a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely [proportional](https://bliptext.com/articles/proportionality-mathematics) to the number of decimal digits of x. A far better estimate of\nπ(x)\nis given by the [offset logarithmic integral](https://bliptext.com/articles/logarithmic-integral-function) function\nLi(x)\n, defined by\nThe [Riemann hypothesis](https://bliptext.com/articles/riemann-hypothesis), one of the oldest open mathematical [conjectures](https://bliptext.com/articles/conjecture), can be stated in terms of comparing\nπ(x)\nand\nLi(x)\n. The [Erdős–Kac theorem](https://bliptext.com/articles/erd-s-kac-theorem) describing the number of distinct [prime factors](https://bliptext.com/articles/prime-factor) also involves the [natural logarithm](https://bliptext.com/articles/natural-logarithm).\nThe logarithm of n [factorial](https://bliptext.com/articles/factorial),\nn! = 1 · 2 · ...\n· n\n, is given by\nThis can be used to obtain [Stirling's formula](https://bliptext.com/articles/stirling-s-formula), an approximation of\nn!\nfor large n.\n\n# Generalizations\n\n## Complex logarithm\n[[File:Complex number illustration multiple arguments.svg|thumb|right|Polar form of\n. Both φ and φ' are arguments of z.|alt=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis.]]\nAll the [complex numbers](https://bliptext.com/articles/complex-number) a that solve the equation\ne^a=z\nare called complex logarithms of z, when z is (considered as) a complex number.\nA complex number is commonly represented as\n, where x and y are real numbers and i is an [imaginary unit](https://bliptext.com/articles/imaginary-unit), the square of which is −1.\nSuch a number can be visualized by a point in the [complex plane](https://bliptext.com/articles/complex-plane), as shown at the right.\nThe [polar form](https://bliptext.com/articles/polar-form) encodes a non-zero complex number z by its [absolute value](https://bliptext.com/articles/absolute-value), that is, the (positive, real) distance r to the [origin](https://bliptext.com/articles/origin-mathematics), and an angle between the real (x) axis \nRe\nand the line passing through both the origin and z. This angle is called the [argument](https://bliptext.com/articles/argument-complex-analysis) of z.\nThe absolute value r of z is given by\nUsing the geometrical interpretation of [sine](https://bliptext.com/articles/sine) and co[sine](https://bliptext.com/articles/sine) and their periodicity in\n2π\n, any complex number z may be denoted as\nfor any integer number k. Evidently the argument of z is not uniquely specified: both φ and\nφ' = φ + 2kπ\nare valid arguments of z for all integers k, because adding\n2kπ\n[radians](https://bliptext.com/articles/radian) or k⋅360° to φ corresponds to \"winding\" around the origin counter-clock-wise by k [turns](https://bliptext.com/articles/turn-geometry).\nThe resulting complex number is always z, as illustrated at the right for\n. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted\nArg(z)\n, with a capital\nA\n, by requiring φ to belong to one, conveniently selected turn, e.g.\n−π < φ ≤ π\nor\n0 ≤ φ < 2π\n. These regions, where the argument of z is uniquely determined are called branches of the argument function.\n[[File:Complex log domain.svg|right|thumb|The principal branch (-π, π) of the complex logarithm,\nLog(z)\n. The black point at\ncorresponds to absolute value zero and brighter colors refer to bigger absolute values.\nThe [hue](https://bliptext.com/articles/hue) of the color encodes the argument of\nLog(z)\n.|alt=A density plot.\nIn the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.]]\n[Euler's formula](https://bliptext.com/articles/euler-s-formula) connects the [trigonometric functions](https://bliptext.com/articles/trigonometric-functions) [sine](https://bliptext.com/articles/sine) and co[sine](https://bliptext.com/articles/sine) to the [complex exponential](https://bliptext.com/articles/complex-exponential):\nUsing this formula, and again the periodicity, the following identities hold:\nwhere\nln(r)\nis the unique real natural logarithm,\nak\ndenote the complex logarithms of z, and k is an arbitrary integer.\nTherefore, the complex logarithms of z, which are all those complex values\nak\nfor which the\nak-th\npower of e equals z, are the infinitely many values\nfor arbitrary integers k.\nTaking k such that\nφ + 2kπ\nis within the defined interval for the principal arguments, then\nak\nis called the principal value of the logarithm, denoted\nLog(z)\n, again with a capital\nL\n. The principal argument of any positive real number x is 0; hence\nLog(x)\nis a real number and equals the real (natural) logarithm.\nHowever, the above formulas for logarithms of products and powers do generalize to the principal value of the complex logarithm.\nThe illustration at the right depicts\nLog(z)\n, confining the arguments of z to the interval.\nThis way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there.\nThis discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding k-value of the continuously neighboring branch.\nSuch a locus is called a [branch cut](https://bliptext.com/articles/branch-cut).\nDropping the range restrictions on the argument makes the relations \"argument of z\", and consequently the \"logarithm of z\", [multi-valued functions](https://bliptext.com/articles/multi-valued-function).\n\n## Inverses of other exponential functions\nExponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm.\nFor example, the [logarithm of a matrix](https://bliptext.com/articles/logarithm-of-a-matrix) is the (multi-valued) inverse function of the [matrix exponential](https://bliptext.com/articles/matrix-exponential).\nAnother example is the p-adic logarithm, the inverse function of the p-adic exponential.\nBoth are defined via Taylor series analogous to the real case.\nIn the context of [differential geometry](https://bliptext.com/articles/differential-geometry), the [exponential map](https://bliptext.com/articles/exponential-map-riemannian-geometry) maps the [tangent space](https://bliptext.com/articles/tangent-space) at a point of a [manifold](https://bliptext.com/articles/differentiable-manifold) to a [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) of that point.\nIts inverse is also called the logarithmic (or log) map.\nIn the context of [finite groups](https://bliptext.com/articles/finite-group) exponentiation is given by repeatedly multiplying one group element b with itself.\nThe [discrete logarithm](https://bliptext.com/articles/discrete-logarithm) is the integer n solving the equation\nb^n = x, where x is an element of the group.\nCarrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups.\nThis asymmetry has important applications in [public key cryptography](https://bliptext.com/articles/public-key-cryptography), such as for example in the [Diffie–Hellman key exchange](https://bliptext.com/articles/diffie-hellman-key-exchange), a routine that allows secure exchanges of [cryptographic](https://bliptext.com/articles/cryptography) keys over unsecured information channels.\n[Zech's logarithm](https://bliptext.com/articles/zech-s-logarithm) is related to the discrete logarithm in the multiplicative group of non-zero elements of a [finite field](https://bliptext.com/articles/finite-field).\nFurther logarithm-like inverse functions include the double logarithm\nln(ln(x))\n, the [super- or hyper-4-logarithm](https://bliptext.com/articles/super-logarithm) (a slight variation of which is called [iterated logarithm](https://bliptext.com/articles/iterated-logarithm) in computer science), the [Lambert W function](https://bliptext.com/articles/lambert-w-function), and the [logit](https://bliptext.com/articles/logit).\nThey are the inverse functions of the [double exponential function](https://bliptext.com/articles/double-exponential-function), [tetration](https://bliptext.com/articles/tetration), of\n, and of the [logistic function](https://bliptext.com/articles/logistic-function), respectively.\n\n## Related concepts\nFrom the perspective of [group theory](https://bliptext.com/articles/group-theory), the identity\nexpresses a [group isomorphism](https://bliptext.com/articles/group-isomorphism) between positive [reals](https://bliptext.com/articles/real-number) under multiplication and [reals](https://bliptext.com/articles/real-number) under addition.\nLogarithmic functions are the only continuous isomorphisms between these groups.\nBy means of that isomorphism, the [Haar measure](https://bliptext.com/articles/haar-measure) ([Lebesgue measure](https://bliptext.com/articles/lebesgue-measure))\ndx\non the reals corresponds to the Haar measure\ndx/x\non the positive reals.\nThe non-negative reals not only have a multiplication, but also have addition, and form a [semiring](https://bliptext.com/articles/semiring), called the probability [semiring](https://bliptext.com/articles/semiring); this is in fact a [semifield](https://bliptext.com/articles/semifield).\nThe logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition ([LogSumExp](https://bliptext.com/articles/logsumexp)), giving an [isomorphism](https://bliptext.com/articles/isomorphism) of semirings between the probability semiring and the [log semiring](https://bliptext.com/articles/log-semiring).\nLogarithmic one-forms\ndf/f\nappear in [complex analysis](https://bliptext.com/articles/complex-analysis) and [algebraic geometry](https://bliptext.com/articles/algebraic-geometry) as [differential forms](https://bliptext.com/articles/differential-form) with logarithmic [poles](https://bliptext.com/articles/pole-complex-analysis).\nThe [polylogarithm](https://bliptext.com/articles/polylogarithm) is the function defined by\nIt is related to the [natural logarithm](https://bliptext.com/articles/natural-logarithm) by\nLi1 (z) = −ln(1 − z)\n. Moreover,\nLis (1)\nequals the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function)\nζ(s)\n.","revisionCount":0},"hasMore":false,"nextCursor":null}