derivative of logarithm with respect to base

The

${\displaystyle \frac{\partial }{\partial a}}{\log }_{a}x=-{\displaystyle \frac{\ln x}{{(\ln a)}^{2}a}}$ (1)

for the partial derivative of logarithm expression with respect to the base $a$ may be derived by denoting first

$${\log }_{a}x=y.$$

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:

$$\frac{\partial y}{\partial a}=\frac{0\ln a-\frac{1}{a}\ln x}{{(\ln a)}^2}=-\frac{\ln x}{{(\ln a)}^{2}a}.$$

Title	derivative of logarithm with respect to base
Canonical name	DerivativeOfLogarithmWithRespectToBase
Date of creation	2013-03-22 19:11:28
Last modified on	2013-03-22 19:11:28
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	4
Author	pahio (2872)
Entry type	Definition
Classification	msc 26-00
Classification	msc 26A09
Classification	msc 26A06
Related topic	Derivative