I know that for $a, b \in \mathbb{R}$, the rule $\log(ab) = \log(a) + \log(b)$ holds.
What about for $a_1, b_1$ in the right half-plane, or $a_2, b_2$ in the sector from $\frac{-3\pi}{4}$ to $\frac{3\pi}{4}$? For the logarithm, take a branch cut on the negative real line.
The logarithm function is $f(x) = \log(x)$ is defined in $f: \mathbb{C} \setminus \mathbb{R}_{-} \mapsto \mathbb{C}$
So, let $z_1 = e^{\rho_1 + i\theta_1}, z_2=e^{\rho_2 + i\theta_2} \in \mathbb{C}$, if $\theta_1 + \theta_2 \neq \pi$, then $$\log(z_1 z_2) = \log(z_1) + \log(z_2)$$