Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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All logarithms of z≠0z\ne0 are Log⁡z+2πik\operatorname{Log}z+2\pi i k, k∈Zk\in\mathbb Z

Statement

For z≠0z\ne0, the solutions of exp⁡w=z\exp w=z are exactly Log⁡z+2πik\operatorname{Log}z+2\pi ik for k∈Zk\in\mathbb 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\ker(\exp)=2\pi i\mathbb Z, and exp⁡z=exp⁡w\exp z=\exp w exactly when z−w∈2πiZz-w\in2\pi i\mathbb Z, The complex exponential maps C\mathbb C onto C∖{0}\mathbb C\setminus\{0\}.

Facts & Assumptions

Given: z≠0z\ne0 and w∈Cw\in\mathbb C.

Proof

technique · direct
1.1

The principal logarithm is one solution by its polar definition.

given
2.1

Equality exp⁡w=exp⁡(Log⁡z)\exp w=\exp(\operatorname{Log}z) is equivalent to w−Log⁡z∈2πiZw-\operatorname{Log}z\in2\pi i\mathbb Z by the fibre theorem.

given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 42 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources