Editing Logarithm (section)

===Change of base===<!-- This section is linked from [[Mathematica]] -->
The logarithm {{math|log<sub>''b''</sub>&thinsp;''x''}} can be computed from the logarithms of {{mvar|x}} and {{mvar|b}} with respect to an arbitrary base&nbsp;{{Mvar|k}} using the following formula:{{refn|group=nb|''Proof:''
Taking the logarithm to base {{mvar|k}} of the defining identity <math display=inline> x = b^{\log_b x},</math> one gets
<math> \log_k x = \log_k \left(b^{\log_b x}\right) = \log_b x \cdot \log_k b.</math>
The formula follows by solving for <math>\log_b x.</math>}}
<math display="block"> \log_b x = \frac{\log_k x}{\log_k b}.</math>

Typical [[scientific calculators]] calculate the logarithms to bases 10 and {{mvar|[[e (mathematical constant)|e]]}}.<ref>{{Citation | last1=Bernstein | first1=Stephen | last2=Bernstein | first2=Ruth | title=Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability | publisher=[[McGraw-Hill]] | location=New York | series=Schaum's outline series | isbn=978-0-07-005023-5 | year=1999 | url=https://archive.org/details/schaumsoutlineof00bern }}, p.&nbsp;21</ref> Logarithms with respect to any base&nbsp;{{mvar|b}} can be determined using either of these two logarithms by the previous formula:
<math display="block"> \log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\log_{e} x}{\log_{e} b}.</math>

Given a number {{mvar|x}} and its logarithm {{math|1=''y'' = log<sub>''b''</sub>&thinsp;''x''}} to an unknown base&nbsp;{{mvar|b}}, the base is given by:

<math display="block"> b = x^\frac{1}{y},</math>

which can be seen from taking the defining equation <math> x = b^{\,\log_b x} = b^y</math> to the power of <math>\tfrac{1}{y}.</math>