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I was playing around with DeMoivre's formula

e

^{i*pi}= -1

and there is something I don't quite understand about taking the natural logarithm of a certain expression. I though that

e

^{2i*pi}= 1

ln[e

^{2i*pi}] = ln (1),

but this yields to an imposibility

2i*pi = 0.

So obviously I am doing something wrong, and when I input ln[e^(2i*pi)] into Wolfram, it gives log[e

^{2i*pi}]=0.

Can anyone explain why Wolfram Alpha translates ln[e^(2i*pi)] to log[e

^{2i*pi}]=0, and why that second expression is true?

Thank you in advance for your time.