Solution verification:$\lim_{x\to 2}\frac{\ln(x-1)}{3^{x-2}-5^{-x+2}}$

This is fine. Here is an alternative approach using taylor series:

First substitute $x-2=y$ to simplify it. Let the required limit be $l$. Then $$l = \lim_{y\to0}\left(\dfrac{\ln(1+y)}{3^y-5^{-y}}\right)$$ $$ = \lim_{y\to0}\left(\dfrac{y-\dfrac{y^2}{2}+\cdots}{(1+y\ln3+\cdots)-(1-y\ln5+\cdots)}\right)$$ $$=\dfrac{1}{\ln3+\ln5}=\dfrac{1}{\ln15}$$