Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02
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All logarithms of z≠0 are Log⁡z+2πik, k∈Z

Statement

For z≠0, the solutions of exp⁡w=z are exactly Log⁡z+2πik for k∈Z. The conventions and prerequisite facts used below are recorded in Complex logarithms, the principal logarithm, and principal and multivalued complex powers, ker⁡(exp⁡)=2πiZ, and exp⁡z=exp⁡w exactly when z−w∈2πiZ, The complex exponential maps C onto C∖{0}.

Facts & Assumptions

Given: z≠0 and w∈C.

Proof

technique · direct
1.1given

The principal logarithm is one solution by its polar definition.

2.1given∎

Equality exp⁡w=exp⁡(Log⁡z) is equivalent to w−Log⁡z∈2πiZ by the fibre theorem.

Depends on

Dependency tree · 2 levels

Sources