Properties of modulus and complex logarithm

Problem

Prove the following properties: a) $\ln(z)+\ln(\bar z)$ has zero imaginary part, while $\ln(z)-\ln(\bar z)$ has zero real part. b) $|zw|=|z|\,|w|$. c) $|z^{2}|=|z|^{2}$.

Solution

a) Since $|\bar z|=|z|$ and $\arg\bar z=-\arg z$: $\ln z=\ln|z|+i\arg z$, $\ln\bar z=\ln|z|-i\arg z$. Hence $\ln z+\ln\bar z=2\ln|z|$ (real, zero imaginary part) and $\ln z-\ln\bar z=2i\arg z$ (purely imaginary, zero real part). b) In exponential form $z=r\,e^{i\theta}$, $w=s\,e^{i\varphi}$, so $zw=rs\,e^{i(\theta+\varphi)}$ and $|zw|=rs=|z|\,|w|$. c) Special case with $w=z$: $|z^{2}|=|z|\,|z|=|z|^{2}$. $\boxed{\,\ln z+\ln\bar z=2\ln|z|\in\mathbb{R},\qquad |zw|=|z||w|\,}$