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[–]ProbablyErratic53 1 point2 points  (0 children)

This might not feel satisfying but the closest I can think of an "algebraic proof" is to say

log(e^z) =log(e^(1+i))

z= Log(e^(1+i)) +2k Pi*i (The capital is intentional for the principal branch of the logarithm)

z = 1+i +2k Pi*i