Interchanging limits and logarithms This is probably not too smart, just wondering of the name of this rule: $$ \log \lim_{x \to x_0}f(x) = \lim_{x \to x_0}\log f(x) $$ A reference to a source and/or proof would be good too.

As long as the function inside the limit symbol is continuous at $x_0$, the statement follows from the continuity of that function. Search for "interchanging limits," and you're sure to find sources.

Edit: Here's something worthwhile from an introductory real analysis textbook.