Is $ \ln |f| $ harmonic? I'd like to show that $\ln |f| $ is harmonic, where $f$ is holomorphic defined on a domain of the complex plane and never takes the value 0.
My idea was to use the fact that $\ln |f(z)| = \operatorname{Log} f(z) - i*\operatorname{Arg}(f(z)) $, but $Log$ is only holomorphic on some part of the complex plane and $\operatorname{Arg}$ is not holomorphic at all.
Any help is welcome!

This is a local result; you need to show that given a $z_0$ with $f(z_0) \neq 0$ there is a neighborhood of $z_0$ on which $\ln|f(z)|$ is harmonic. 
Fix $z_0$ with $f(z_0) \neq 0$. Let $\log(z)$ denote an analytic branch of the logarithm defined on a neighborhood of $f(z_0)$. Then the real part of $\log(z)$ is $\ln|z|$; any two branches of the logarithm differ by an integer multiple of $2\pi i$. The function $\log(f(z))$, being the composition of analytic functions, is analytic on a neighborhood of $z_0$. The real part of this function is $\ln|f(z)|$, which is therefore harmonic on a neighborhood of $z_0$, which is what you need to show.
