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

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.

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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.

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Thank you Zev. Actually I found this relation in an algorithm performance data. The x represents the input scale. The f(x) represents the running time. Then I want know if there is a mathematical relation between them. Any suggestion? –  Kai Mar 1 '13 at 4:40
My argument above demonstrates that $f(x)$ would have to be 0 anywhere it even makes sense to talk about it. Perhaps you copied down the expression incorrectly? –  Zev Chonoles Mar 1 '13 at 4:46
I think Kai meant it for specific numbers $x_1$ and $x_2$.
In that case there may be such function $f(x)$ which would have the desired form. It would just mean that there is an intersection of function $g(x_1,x_2)=\frac{f(x_1)}{f(x_2)}$ with $h(x_1,x_2)=\log\frac{x_1}{x_2}$.