Editing Logarithm (section)

==Definition==
Given a positive [[real number]] {{mvar|b}} such that {{math|''b'' ≠ 1}}, the ''logarithm'' of a positive real number {{mvar|x}} with respect to base&nbsp;{{mvar|b}}{{refn|The restrictions on {{mvar|x}} and {{mvar|b}} are explained in the section [[#Analytic properties|"Analytic properties"]].|group=nb}} is the exponent by which {{mvar|b}} must be raised to yield {{mvar|x}}. In other words, the logarithm of {{mvar|x}} to base&nbsp;{{mvar|b}} is the unique real number&nbsp;{{mvar|y}} such that <math>b^y = x</math>.<ref>{{Citation|last1=Kate|first1=S.K.|last2=Bhapkar|first2=H.R.|title=Basics Of Mathematics|location=Pune|publisher=Technical Publications|isbn=978-81-8431-755-8|year=2009 }}, chapter 1</ref>

The logarithm is denoted "{{math|log<sub>''b''</sub>&thinsp;''x''}}" (pronounced as "the logarithm of {{mvar|x}} to base&nbsp;{{mvar|b}}", "the {{nowrap|base-{{mvar|b}}}} logarithm of {{mvar|x}}", or most commonly "the log, base&nbsp;{{mvar|b}}, of {{mvar|x}}").

An equivalent and more succinct definition is that the function {{math|log<sub>''b''</sub>}} is the [[inverse function]] to the function <math>x\mapsto b^x</math>.

===Examples===
* {{math|1=log<sub>2</sub>&thinsp;16 = 4}}, since {{math|1=2<sup>4</sup> = 2 × 2 × 2 × 2 = 16}}.
* Logarithms can also be negative: <math display="inline">\log_2 \! \frac{1}{2} = -1</math> since <math display="inline">2^{-1} = \frac{1}{2^1} = \frac{1}{2}.</math>
* {{math|log<sub>10</sub>&thinsp;150}} is approximately 2.176, which lies between 2 and 3, just as 150 lies between {{math|1=10<sup>2</sup> = 100}} and {{math|1=10<sup>3</sup> = 1000}}.
* For any base&nbsp;{{mvar|b}}, {{math|1=log<sub>''b''</sub>&thinsp;''b'' = 1}} and {{math|1= log<sub>''b''</sub>&thinsp;1 = 0}}, since {{math|1=''b''<sup>1</sup> = {{mvar|b}}}} and {{math|1=''b''<sup>0</sup> = 1}}, respectively.