TITLE: How do you prove that $\ln|f(z)|$ is harmonic?
QUESTION [13 upvotes]: Suppose that $f(z)$ is analytic and nonzero in a domain $D$.  Prove that $\ln|f(z)|$ is harmonic in $D$. 
I know the laplacian equation but I'm not sure how to use it.

REPLY [14 votes]: EDIT: As 5pm points out below, this is of course actually good enough since every domain in $\mathbb{C}$ is locally simply connected and harmonicity is a local property!
This was too long for a comment, but I thought it might be nice to know.
There is a much nicer, less computational way to prove this result if we assume further that $D$ is simply connected. In particular, recall that if $D$ is a simply connected domain in $\mathbb{C}$ and $h$ a non-vanishing holomorphic function on $D$ then $h=e^g$ for some holomorphic function $g$ (this is because we can define a branch of the logarithm on $D$). 
So, if $D$ was simply connected we'd know that $f=e^g$ for some holomorphic $g$, and then 
$$\log|f|=\log|e^g|=\log(\exp(\text{Re}(g))=\text{Re}(g)$$
and since $g$ is harmonic (since $g$ was holomorphic!) we're done.