Why aren't any derivatives defined to measure logarithmic change in a function?

Question

For example, is this derivative useless: $$\lim_{h\rightarrow 0}\log_{\frac{x+h}{x}}\frac{f(x+h)}{f(x)}$$ It evaluates to $f^{'}(x)\cdot \frac{x}{f(x)}$, where $f^{'}(x)$ is the normal derivative of the function. I could think of another one: $$\lim_{h\rightarrow 0}\left(\frac{f(x+h)}{f(x)}\right)^{\frac{1}{h}}$$ This one evaluates to $e^{\frac{f^{'}(x)}{f(x)}}$ I had used these in approximations in some of my previous posts. But that's not a good use to talk about when there are so many other methods for that. Can you think of any other use of these?