The

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

(1) 
for the partial derivative^{} of logarithm^{} expression with respect to the base $a$ may be derived by denoting first

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

By the definition of logarithm, this equation means the same as
where we can take the natural logarithms^{}

$$y\mathrm{ln}a=\mathrm{ln}x$$ 

solving then

$$y=\frac{\mathrm{ln}x}{\mathrm{ln}a}.$$ 

Then, the differentiation^{} is easy:

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