Relationship between logarithm (multivalued) and positive integer exponents

Theorem

Let $z \in \mathbb{C}$ and $n$ be a positive integer. Then the following formula holds: $$\mathrm{Log} \left( z^n \right) \subset n \mathrm{Log}(z),$$ where $\mathrm{Log}$ denotes the logarithm (multivalued) and $n \mathrm{Log}(z)=\left\{nw \colon w \in \mathrm{Log}(z)\right\}$.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.10$