I encountered this proof durring my self studing and it really makes my life harder. This is function $f$ $\colon \mathbb R \to \mathbb R$ which derivative at point $a$, such that $f(a) > 0$.
I need to prove that : $\lim_{x\to 0}( \frac{f(x)}{f(a)})^\frac{1}{ lnx - lna} = \frac{f'(a)}{f(a)}\cdot a$
So far I found that the left side equal to $1$ (only if $\ln a$ is defined which means $a>0$). for the other side I tried to break it into the definition of derivation by $\lim$ but no success so far.
I would like to hear some good idea how to continue from here.
Hint $$ \left(\frac{f\left(x\right)}{f\left(a\right)}\right)^{1/(\ln(x)-\ln(a))}=\text{exp}\left(\frac{\ln\left(f\left(x\right)\right)-\ln\left(f\left(a\right)\right)}{\ln(x)-\ln(a)}\right) $$