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:

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