How do you determine the "period" of a complex logarithm as a multivalued function in an arbitrary (real or complex) base?
I apologize in advance if my terminology is incorrect, but let me illustrate what I'm talking about. Take $log_e(z)$ for any (non-zero) complex number $z$:
- $log_e(-1) = \pi i + n2\pi{i}$ for any integer $n$
- $log_e(2 + 4i) = 1.498 + 1.107 + n2\pi{i}$ for any integer $n$
So $log_e(z)$ is periodic over $2\pi{i}$. I understand how this arises with $log_e$, due to $z = re^{iθ}$, but I don't know how to determine this.
Specifically: how do I figure this out for an arbitrary base $b$ in $log_b(z)$? Is it easier if $b$ is a positive number greater than 1? What about when $b$ is an arbitrary complex number?
I have tried starting with $log_b(z) = log_e(z)/log_e(b)$ and expanding everything out and sort of get an answer by playing around with the results and testing against numeric computations on the computer, but I feel like there has to be a nice, simple, closed form equation to compute this. I'm probably missing something obvious, but I don't know what it is.
EDIT: Revisiting my attempts, what I have so far:
- Given: $z = re^{iθ}$, $log_e(z) = log_e(r) + iθ + n2\pi{i}$ $\forall n \in Z$
- Then: $log_b(z) = log_e(z)/log_e(b) = (log_e(r) + iθ)/log_e(b) + (n2\pi{i})/log_e(b)$ $\forall n \in Z$
- The periodicity would be the last term above $2\pi{i}/log_e(b)$. This didn't seem to always be right ... but see my answer below.