Conditions for being harmonic in a region $R$ 
Prove that the function $G=\ln|f(z)|$ is harmonic in a region $R$ if $f(z)$ is analytic in $R$ and also $f(z)\cdot f'(z)$ does not equal zero in $R$.

My difficulty here is that the expression for the Laplacian of $G$ is very big and ugly, and I know that I have to apply the Cauchy-Riemann Equations somewhere , but it is not clear to me how and where.
Also the condition of the multiplication of the complex function with its derivative not being zero looks rather mysterious . Any help would be appreciated.

Trigger warning: If the phrase "multiple-valued function" causes you pain, then stop reading here.
Suppose we define
$$
\operatorname{Log} z = \log |z| + i\arg z \tag 1
$$
where $\operatorname{Log}$ is "multiple-valued" because $\arg$ is multiple-valued.  The different branches of $\operatorname{Log}$ differ by a constant and so all have the same derivative, which is $1/z$.  If you like, just restrict attention to some region within which $\operatorname{Log}$ is single-valued.  Then it's holomorphic, and thus harmonic.  If $f$ is holomorphic then the chain rule tells us $\operatorname{Log}\circ f$ is holomorphic.
So suppose you can prove


*

*$(1)$ is holomorphic (maybe by showing that it's locally an inverse of $\exp$?), and

*The real part of a holomorphic function is harmonic.


Then you've got it.  If your question is how to prove these two points, then I would need to add more, but otherwise maybe I can hope this answers your question.
