$\frac{f(x_1)}{f(x_2)} = \log(\frac{x_1}{x_2}) \implies f(x)=\;?$

Question

If $$\frac{f(x_1)}{f(x_2)} = \log\left(\frac{x_1}{x_2}\right),$$

what is $f(x)$? I mean the simplest form of $f(x)$, and what math technique you use to solve this problem? Thanks.

Answer 1

For any value of $x$, if $f(x)\neq 0$, then we have that $$1=\frac{f(x)}{f(x)}=\log\left(\frac{x}{x}\right)=\log(1)=0,$$ which is impossible. Therefore, anywhere such a function is even defined, it will have to be 0.