Logarithm of a quotient is a difference of logarithms

Theorem

Let $z_1, z_2 \in \mathbb{C} \setminus (-\infty,0]$ with $z_2 \neq 0$ and $- \pi < \mathrm{arg}(z_1) - \mathrm{arg}(z_2) \leq \pi$. Then the following formula holds: $$\log \left( \dfrac{z_1}{z_2} \right) = \log(z_1) - \log(z_2),$$ where $\mathrm{arg}$ denotes the argument and $\log$ denotes the logarithm.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.9