When does $\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$?

Question

When does $$\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$$

I have seen different things from different sources. For example, this and this. Does $f(x)$ have to be continuous at c, does $\log (f(x))$ have to be continuous at c, or something else?

Answer 1

If $\lim_{x\rightarrow c}f(x)> 0,$ then it is OK, i.e, Logrithmic function need to be continuous at $\lim_{x\rightarrow c}f(x).$