The symbol "Log" denotes the complex logarithm.
Let $w$ be a complex number so that $w = u+iv$ for some reals $u, v.$ We have $$\mbox{Log}(z^{w}) = \log |z^{w}| + i\arg (z^{w}) = u\log |z| - v\arg (z) + i[v\log |z| + u\arg (z)] + i\arg(z^{w}).$$ Then I am not sure how to proceed?
On
It all depends on the meaning of $\operatorname{Log}$ and $z^w$.
Say that $\operatorname{Log}(z):=\{z' \in \mathbb{C}|\exp(z')=z\}$ is a subset of $\mathbb{C}$. Then for any $w$ you can define the set $$ z^w := \{\exp(z' w) | z' \in \operatorname{Log}(z) \} $$ so that the only good definition of $\operatorname{Log}(z^w)$ might be $$ \{z'' | \exp(z'') \in z^w\}=\{z''| \exp(z'')=\exp(z'w) \text{ for some } z' \text{ such that } \exp{z'}=z \} $$
If $B(z,w):=\{wz'+2\pi i n | z' \in \operatorname{Log}(z)\}$ then the equality of sets $$ B(z,w)=\operatorname{Log}(z^w) $$ is straight forward.
$ln(z^w)=ln(|z^w|)+i\arg(z^w)$ where $-\pi < \arg(z^w) \leq \pi$.
How can we calculate $|z^w|$? Say $z=re^{i\theta}$, then $z^w=r^we^{w\theta i}$ Example: $|i^i|=|e^{i.i\pi/2}|=|e^{-\pi/2}|=e^{-\pi/2}$.
How can we calculate $arg(z^w)$? in same way.. $arg(i^i)=0$