Keep the domain in view.

The derivative of $\ln x$ is $\frac{1}{x}$ on $x>0$. Applying the chain rule gives $\frac{g'(x)}{g(x)}$ for $\ln(g(x))$, wherever $g(x)$ is positive. State this real domain alongside a derivative.

Because ln and exp are inverse functions, $\ln(e^x)$ = x and $e^{\ln x}$ = x for $x>0$. Taking logarithms solves exponential equations. The derivative measures local sensitivity: equal absolute input changes have larger effects when x is small.