Can $\ln$ be moved out of a limit, similar to an exponent?
Limit of composite functions: If $\lim\limits_{x \to a} f(x) = L$ and $g$ is continuous at $x = L$ then $\lim\limits_{x \to a} g(f(x))= g \left( \lim\limits_{x \to a} f(x) \right)$.
In words, that simply means you can move $\lim\limits_{x \to a}$ in and out of $g$ if $g$ is continuous at $\lim\limits_{x \to a} f(x)$.
Since $\ln ( \cdot )$ is continuous everywhere on $(0, \infty)$, you can move the $\lim\limits_{x \to a}$ in and out of $\ln( \cdot )$ whenever you feel like, as long as the $\lim\limits_{x \to a } f(x)$ is positive, of course.