Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$

Consider the function $\log(1-a\,z)$ and think mean value.

Hint:

Note that $\log|1-ae^{i\theta}|$ is the real part of $\log(1-ae^{i\theta})$. Then try differentiating with respect to $a$. Then notice that integrating around the unit circle $$ \frac1i\oint\frac{\mathrm{d}z}{1-az}=0 $$ when $|a|<1$.