I can kind of see whats going on, but not well enough to construct a logical progression towards a proof. Any ideas?
That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say $\displaystyle \exp(f(z)) = e^{u+iv} = e^ue^{iv}$ and therefore $\displaystyle |\exp(f(z))| = e^u$ (since $\displaystyle e^u>0$ and $\displaystyle |e^{iv}|=1$). Then apply Liouville's theorem.