DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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 $x > 0$ .

Definition references

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

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 $R$ onto $(0, \infty)$

The natural logarithm as the inverse of the exponential function

Definition

Definition references

Remarks

Depends on

Dependency tree · 2 levels

Sources