26-00-DerivativeOfLogarithmWithRespectToBase.tex

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\pmcanonicalname{DerivativeOfLogarithmWithRespectToBase}
\pmcreated{2013-03-22 19:11:28}
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\pmtitle{derivative of logarithm with respect to base}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{26-00}
\pmclassification{msc}{26A09}
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\pmrelated{Derivative}

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\begin{document}
The \PMlinkescapetext{formula}
\begin{align}
\frac{\partial}{\partial a}\log_ax \;=\; -\frac{\ln{x}}{(\ln{a})^2a}
\end{align}
for the partial derivative of logarithm expression with respect to the base $a$ may be derived by denoting first
$$\log_ax \;=\; y.$$
By the definition of logarithm, this equation means the same as
$$a^y \;=\; x,$$
where we can take the natural logarithms
$$y\ln{a} \;=\; \ln{x}$$
solving then
$$y \;=\; \frac{\ln{x}}{\ln{a}}.$$
Then, the differentiation is easy:
$$\frac{\partial y}{\partial a} \;=\; \frac{0\ln{a}-\frac{1}{a}\ln{x}}{(\ln{a})^2} \;=\; -\frac{\ln{x}}{(\ln{a})^2a}.$$
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