TITLE: Logarithm of absolute value of a holomorphic function harmonic?
QUESTION [15 upvotes]: Let $f:U\rightarrow\mathbb{C}$ be holomorphic on some open domain $U\subset\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ and $f(z)\not=0$ for $z\in U$.

Is it true that $z\mapsto \log(|f(z)|)$ is harmonic on $U$ ?

I guess the answer is yes and if that is true, how can I see that without a long and nasty calculation?

REPLY [7 votes]: Locally (but not necessarily globally), $\log f(z)$ is an analytic function because $\log$ is an analytic function.  The real part of $\log f(z)$ is $\log |f(z)|$, i.e. the polar representation of the complex number $w$ is  $w = r e^{i\theta}$ where $r = |w|$, and 
$\log w = \log r + i \theta$ so $\text{Re}(\log w) = \log r = \log |w|$.  The real part of an analytic function is harmonic.