Editing Logarithm (section)

{{short description|Mathematical function, inverse of an exponential function}}
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[[File:Logarithm plots.png|right|thumb|upright=1.35|Plots of logarithm functions, with three commonly used bases. The special points {{math|log<sub>''b''</sub>&thinsp;''b'' {{=}} 1}} are indicated by dotted lines, and all curves intersect in {{math|1=log<sub>''b''</sub>&thinsp;1 = 0}}.]]

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In [[mathematics]], the '''logarithm''' of a number is the [[exponent]] by which another fixed value, the '''base''', must be raised to produce that number. For example, the logarithm of {{math|1000}} to base {{math|10}} is {{math|3}}, because {{math|1000}} is {{math|10}} to the {{math|3}}rd power: {{math|1000 {{=}} 10{{sup|3}} {{=}} 10 × 10 × 10}}. More generally, if {{math|''x'' {{=}} ''b''{{sup|''y''}}}}, then {{mvar|y}} is the logarithm of {{mvar|x}} to base {{mvar|b}}, written {{math|log{{sub|''b''}} ''x''}}, so {{math|log{{sub|10}} 1000 {{=}} 3}}. As a single-variable function, the logarithm to base {{mvar|b}} is the [[inverse function|inverse]] of [[exponentiation]] with base {{mvar|b}}.

The logarithm base {{math|10}} is called the ''decimal'' or [[common logarithm|''common'' logarithm]] and is commonly used in science and engineering. The [[natural logarithm|''natural'' logarithm]] has the number&nbsp;[[e (mathematical constant)|{{math|''e'' ≈ 2.718}}]] as its base; its use is widespread in mathematics and [[physics]] because of its very simple [[derivative]]. The [[binary logarithm|''binary'' logarithm]] uses base {{math|2}} and is widely used in [[computer science]], [[information theory]], [[music theory]], and [[photography]]. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written {{math|log&thinsp;''x''}}.

Logarithms were introduced by [[John Napier]] in 1614 as a means of simplifying calculations.<ref>{{citation |url=http://archive.org/details/johnnapierinvent00hobsiala |title=John Napier and the invention of logarithms, 1614; a lecture |last=Hobson |first=Ernest William |date=1914 |publisher=Cambridge University Press }}</ref> They were rapidly adopted by [[navigator]]s, scientists, engineers, [[surveying|surveyors]], and others to perform high-accuracy computations more easily. Using [[logarithm table]]s, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a [[product (mathematics)|product]] is the [[summation|sum]] of the logarithms of the factors:
<math display="block"> \log_b(xy) = \log_b x + \log_b y,</math>
provided that {{mvar|b}}, {{mvar|x}} and {{mvar|y}} are all positive and {{math|''b'' ≠ 1}}. The [[slide rule]], also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from [[Leonhard Euler]], who connected them to the [[exponential function]] in the 18th century, and who also introduced the letter {{mvar|e}} as the base of natural logarithms.<ref>{{citation|title=Theory of complex functions|last=Remmert, Reinhold.|date=1991|publisher=Springer-Verlag|isbn=0387971955|location=New York|oclc=21118309}}</ref>

[[Logarithmic scale]]s reduce wide-ranging quantities to smaller scopes. For example, the [[decibel]] (dB) is a [[Units of measurement|unit]] used to express [[Level (logarithmic quantity)|ratio as logarithms]], mostly for signal power and amplitude (of which [[sound pressure]] is a common example). In chemistry, [[pH]] is a logarithmic measure for the [[acid]]ity of an [[aqueous solution]]. Logarithms are commonplace in scientific [[formula]]e, and in measurements of the [[Computational complexity theory|complexity of algorithms]] and of geometric objects called [[fractal]]s. They help to describe [[frequency]] ratios of [[Interval (music)|musical intervals]], appear in formulas counting [[prime number]]s or [[Stirling's approximation|approximating]] [[factorial]]s, inform some models in [[psychophysics]], and can aid in [[forensic accounting]].

The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the [[complex logarithm]] is the multi-valued [[inverse function|inverse]] of the complex exponential function. Similarly, the [[discrete logarithm]] is the multi-valued inverse of the exponential function in finite groups; it has uses in [[public-key cryptography]].