TITLE: Is $ \ln |f| $ harmonic?
QUESTION [8 upvotes]: 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!

REPLY [6 votes]: Hint:  Write $f = u + iv$ where $u,v$ are real functions on the plane.  Now your function is $\frac12 \ln (u^2 + v^2)$.  Now compute directly the laplacian of this and make sure to use at some point that $u$ and $v$ satisfy the Cauchy-Riemann equations....