Logarithm Via Multiplication By Some Function

Question

This question may be totally rubbish, I'm not sure...

Basically, I'm wondering if I have some expression, say, $$f(z)A(z)$$ where $A(z)$ is some arbitraryly changing function in $z$ totally out of my control, is there are function, $f(z)$, by which I can multiply $A(z)$, which results in me taking the logarithm of $A(z)$?

I have a feeling the answer is simply no... and this is a really silly question, but just wanting to make sure.

Thanks.

Answer 1

How about defining $f(z) = {Log(A(z)) \over A(z)}$ for $A(z) \in \mathbb{R}^+$ (not positive $A(z)$ does not makes sense, since you want $log(A(z))$ to exist.

EDIT:
Since author requires $f(z)$ to work for any $A(z)$, the answer is simply no. There is no constant $c$ such that $c\cdot A(z)= log(A(z))$ for general $A(z)$