Different domains for (apparently) equivalent functions

Question

Let's look at:

$f_1(x)=ln(x^2-4)$

$f_2(x)=ln(x-2)+ln(x+2)$

Every high school student can tell they are the same, but the first is defined only for $\{x<-2\}\cup\{x>2\}$, and the latter is defined only for $\{x>2\}$.

Do I miss something? I thought $ln(xy)=ln(x)+ln(y)$ is always true...

Answer 1

These are different functions since they have different domain, but they have the same values on their common domain. And the law $ln(xy)=ln(x)+ln(y)$ is true if and only if $x,y>0$.