If $f(z) = u(x, y)+ iv(x, y)$ is an analytic function in a domain $D$ and $f(z) \ne 0$ for all $z \in D$, show that $φ(x, y) = \ln |f(z)|$ is harmonic in $D$.
The above question was taken from Dennis Zill Complex Analysis Chapter 3.3 any help on demonstrating it would be great.
Let $g(z) = \log z$.
$$g(f(z)) = \log f(z) = \ln |f(z)| + i \arg f(z)$$
Since $f(z) \ne 0$, there is a neighborhood around $f(z)$ away from the origin and we can take a branch of the logarithm that does not go through this neighborhood.
So $g(f(z))$ is holomorphic in this neighborhood, and $\ln |f(z)|$ is the real part of a holomorphic function, thus it is harmonic.