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In [[mathematics]], the '''logarithm''' is the [[inverse function]] to [[exponentiation]]. That means that the logarithm of a number {{mvar|x}} to the '''base''' {{mvar|b}} is the [[exponent]] to which {{mvar|b}} must be raised to produce {{mvar|x}}. For example, since {{math|1000 {{=}} 10<sup>3</sup>}}, the ''logarithm base'' <math>10</math> of {{math|1000}} is {{math|3}}, or {{math|log<sub>10</sub> (1000) {{=}} 3}}. The logarithm of {{mvar|x}} to ''base'' {{mvar|b}} is denoted as {{math|log<sub>''b''</sub> (''x'')}}, or without parentheses, {{math|log<sub>''b''</sub> ''x''}}. When the base is clear from the context or is irrelevant it is sometimes written {{math|log ''x''}}.
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 [[e (mathematical constant)|{{math|''e'' ≈ 2.718}}]] as its base; its use is widespread in mathematics and [[physics]]
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|others=University of California Libraries}}</ref> They were rapidly adopted by
<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>
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