Q:

Explanation of proof derivative of logarithms

Why does the proof of the derivative of $log_a(x)$ involve natural logarithms like $ln$? Couldn't creators of the proof used non-natural logarithms like $log$ instead? What is the reasoning behind it?

A:

I don't know if this is what you are looking for, but consider this
$$
\frac{\log_a(x + h) -\log_a(x)}{h} = \frac{1}{h}\log_a\left(1 + \frac{h}{x}\right) = \log_a\left(1 + \frac{h}{x}\right)^{1/h}
$$
In the limit when $h$ goes to $0$ you have
$$
\frac{{\rm d}}{{\rm d}x}\log_ax = \lim_{h\to 0}\frac{\log_a(x + h) -\log_a(x)}{h} = \log_a \lim_{h\to 0}\left(1 + \frac{h}{x}\right)^{1/h} = \log_a e^{1/x}
$$
The problem is now how to evaluate this last expression, let us call 
$$
y = \log_a e^{1/x}
$$
This is equivalent to
$$
a^y = e^{1/x}
$$
You know that the inverse of $e$ is $\ln$, so
$$
\ln a^y = \frac{1}{x}
$$
or equivalently
$$
y = \frac{1}{x \ln a} = \frac{{\rm d}}{{\rm d}x}\log_ax
$$
The appearance of the natural logarithm in this expression is the a natural consequence (pun not intended) of $e$ showing up in the limit. Note that at no point a relation between $\log_a$ and $\ln$ was assumed.

