Logarithm of product is a sum of logarithms

Theorem

Let $z_1,z_2 \in \mathbb{C}$ such that $-\pi < \mathrm{arg}(z_1) + \mathrm{arg}(z_2) \leq \pi$. Then the following formula holds: $$\log(z_1z_2) = \log(z_1)+\log(z_2),$$ where $\log$ denotes the logarithm.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.7