How to Prove That Branch Points Of $\log (f(z))$ are Zeros and Poles of $f(z)$?

Question

In many books and notes, It is written that the branch points of $\log (f(z))$ are points $z\in \Bbb C$ such that $f(z)=0$ or $z$ is a pole of $f$. Here we assuming that zero of $f$ is the sense of analytic functions. I am wondering how to prove this. Any help would be appreciated. Also, are there related theorems to find branch points of arbitrary complex functions?

Thanks in advance!