Partial derivative with respect to y of logarithm?

Question

I don't understand how to do partial derivatives of log.

My question is how to do $\frac{\partial}{\partial y} (\log_y(9x))$. Do I treat $9x$ as a constant? I know the general derivative of a log is $\frac 1{x\ln(b)}$. Please help?

Answer 1

$9x$ would be a constant. What you can do is let $f(x,y)=\log_y(9x)$. Then using change of base, $f(x,y)=\frac{\ln(9x)}{\ln(y)}$. Then $f_y=\frac{\ln(y)0-\ln(9x)\frac{1}{y}}{\ln^2(y)}=\frac{-\ln(9x)}{y\ln^2(y)}$

Edit: I interpreted the post to mean log base $y$, others might have interpreted differently.

Answer 2

Yes, $9x$ is a cosntant considered as a function in the variable $y$.