What limits are there on the property that $a \space \ln(i) = \ln(i^a) $ For what values does the property $a \space \ln(i) = \ln(i^a) $ hold? I found to my dismay that $ 4 \ln(i) $ was not returning the same results as $ \ln(i^4) = 0$. In addition, does there exist a log base where if $a$ is a multiple of $4$, and a base $b$ is chosen, then  $ a \space \log_b(i) = \log_b(i^4) = 0$? Alternatively, is there another function $f$ of the form $a \space f(i) = f(i^a) $ that exists? I would be eager to know about either option.
Thank you in advance.
 A: When we write a function $z \mapsto b^z$ in the context of complex analysis, we mean $$z \mapsto \exp(z \log b) ,$$ but as you know $\log$ has infinitely many branches (owing to the fact that $\exp$ is periodic), and so our function depends on our choice of branch for $\log$. Correspondingly, so do the values for which, e.g., $\log(i^a) = a \log i$ holds.
Example Consider the branch of $\log$ with imaginary part in $\left[0, 2 \pi \right)$. Then $\log i = \frac{\pi i}{2}$, so $4 \log i = 2 \pi i$, which is not in the codomain of our branch, hence we cannot have $4 \log i = \log(i^4)$. On the other hand, for real $a$ such that $\Im(a \log i) \in [0, 2 \pi)$, that is, $a \in [0, 4)$, the identity $\log(i^a) = a \log i$ does hold.
A: The property $a\ln(i) = \ln(i^a)$ or more generally, $a\ln(z) = \ln(z^a)$ for any $z\in\mathbb C$, is true only for the Principal Branch of the Complex Logarithm.
This is due to the fact that the  complex Logarithm is a multi-valued function. In $\mathbb C$, a branch of the (complex) logarithm is generally defined as
$$\log z=\{\ln|z|+i\arg(z)+2\pi ik\ :\ k\in\mathbb Z \}.$$
Clearly, this is neither analytic nor even continuous at 0. To get rid of this problem and obtain an analytic branch of the logarithm, we often use
$$\operatorname{Log} z:= \ln|z|+i\operatorname{Arg}(z),$$ where $\operatorname{Arg}(\cdot)$ denotes the Principal Argument. This branch is commonly called the Principal Branch.
It is this branch of the logarithm that satisfies all nice properties of the usual (real) logarithm such as:

*

*$\operatorname{Log}(zw) =\operatorname{Log}z + \operatorname{Log}w $.

*$\operatorname{Log}(z^a)=a\operatorname{Log}(z)$. Etc.

