the question is whether $$ \log\left(\frac{z_1}{z_2}\right)=\log(z_1) - \log(z_2) $$ holds for any non-zero complex numbers $z_1$ and $z_2$.
Here, $\log$ is the principal value of the logarithm, with definition $$ \log(z) = \ln|z| + i \text{Arg}z $$
I have tried to substitute $z_1$ and $z_2$ by their polar forms, $r_1 e^{i \theta_1}$ and $r_2 e^{i \theta_2}$ respectively, but then I get to
$$ \ln(r_1) -\ln(r_2) + i^{\theta_1 - \theta_2} $$
which is not what i want. Any help?