Prove some properties of the complex logarithm

Extend the logarithm function to all nonzero complex numbers $z$ by defining

$\operatorname{Log} z = \log |z| + i \arg (z).$

Use this formula to prove the following properties of the complex logarithm.

$\operatorname{Log} (-1) = \pi i$ , $\displaystyle{\operatorname{Log} (i) = \frac{\pi i}{2}}$ .
$\operatorname{Log} (z_1 z_2) = \operatorname{Log} z_1 + \operatorname{Log} z_2 + 2n \pi i$ for $n$ an integer.
$\displaystyle{\operatorname{Log} \left( \frac{z_1}{z_2} \right) = \operatorname{Log} z_1 - \operatorname{Log} z_2 + 2n \pi i}$ , where $n$ is an integer.
$e^{\operatorname{Log} z} = z$ .

Proof. For these we use the definition and compute,
$\operatorname{Log} (-1) = \log |-1| + i \arg (-1) = 0 + \pi i = \pi i,$

and

$\operatorname{Log} (i) = \log |i| + i \arg(i) = 0 + \frac{\pi i}{2} = \frac{1}{2} \pi i. \qquad \blacksquare$
Proof. Let $z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$ . Then,
$\begin{align*} \operatorname{Log} (z_1 z_2) &= \log |z_1 z_2| + i \arg(z_1 z_2) \\ &= \log (|z_1||z_2|) + i \arg (\theta_1 + \theta_2 + 2n \pi) \\ &= \log |z_1| + i \theta_1 + \log |z_2| + i \theta_2 + 2n i \pi \\ &= \operatorname{Log} z_1 + \operatorname{Log} z_2 + 2n i \pi. \qquad \blacksquare \end{align*}$
Proof. Again, we compute,
$\begin{align*} \operatorname{Log} \left( \frac{z_1}{z_2} \right) &= \log \left| \frac{z_1}{z_2} \right| + i \arg \left( \frac{z_1}{z_2} \right) \\[9pt] &= \log \frac{|z_1|}{|z_2|} + i (\theta_1 - \theta_2 + 2n i \pi ) \\[9pt] &= \log |z_1| + i \theta_1 - \log|z_2| - i \theta_2 + 2n i \pi \\[9pt] &= \operatorname{Log} z_1 - \operatorname{Log} z_2 + 2n i \pi. \qquad \blacksquare \end{align*}$
Proof. Finally, we have
$e^{\operatorname{Log} z} = e^{\log |z| + i \arg(z)} = e^{\log |z|}e^{i \arg (z)} = e^{\log r} e^{i \theta} = re^{i \theta} = z. \qquad \blacksquare$