I'm trying to show that if $f$ is a harmonic function, then so is $\log|f|$. Moreover, I'm trying to do this using the following operator:
$$ \Delta = 4\frac{\partial}{\partial z} \frac{\partial}{\partial \overline{z}} $$
rather than using another method. Could anyone show me how to do this computation?
You should assume that $f$ is holomorphic; harmonic is not enough (for example, $\log|\operatorname{Re} z |$ is not harmonic).
Formaul computation is not too bad: $$\log|f| = \frac12 \log(f\bar f) = \frac12\log f+\log \bar f \tag1$$ and each term on the right is killed by one of two derivatives you have in the formula for $\Delta$. Some effort must be spent on explaining the meaning of (1). Logarithm can be taken in a neighborhood of $z$ provided $f(z)\ne 0$. We can use whatever branch we want, because derivatives kill constants anyway.
At the points where $f(z)=0$, the function $\log|f|$ is not harmonic.