DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The natural logarithm as the inverse of the exponential function

Definition

For $x > 0$ , define $\log x$ to be the unique real $y$ such that $\exp (y) = x$ . This is well-defined because The exponential is a continuous bijection from $R$ onto $(0, \infty)$ states that $\exp : R \to (0, \infty)$ is a bijection.

Thus $\log : (0, \infty) \to R$ is the inverse function of $\exp$ . In particular, $\log (\exp y) = y$ for every $y \in R$ and $\exp (\log x) = x$ for every $x > 0$ .

Remarks

The domain is exactly $(0, \infty)$ . This definition assigns no real logarithm to $0$ or to a negative number.

Depends on

The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\infty)$

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 8 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

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)