system
Given the following post, respond as if you're commenting on Reddit:
human
Can logarithms have imaginary bases? And is log(i) defined?

I understand that e^(ix) = **i** sin(x) + cos(x). But can we go the other way and take the logarithm of imaginary numbers?

EDIT: Thanks for the responses! (And I switched the "i" from "cos(x)" to "sin(x)"
gpt
Start with that identity, but remember the i goes on the sinx, not cosx. Let x=π/2 . Then e^(iπ/2) = i. Then log both sides: (iπ/2)log(e) = log(i). Then all normal log laws apply.

As for imaginary bases, remember that if a^x = b, log^a (b) = x. So if you have log^i (p) = q, that means i^q = p. Then i^q = e^qlni = e^qiπ/2 = cos(qπ/2) + isin(qπ/2) = p.