CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-13

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 integral logarithm $L$ is the published natural logarithm

Statement

For every $x > 0$ ,

$L (x) = \log x .$

Facts & Assumptions

Given: $x > 0$ .

[F1]

$E = L^{- 1}$ (The integral exponential $E : R \to (0, \infty)$ as the inverse of $L$ ).

[L1]

$E = \exp$ on $R$ (The integral exponential $E$ is the published exponential function).

[F2]

The natural logarithm is the inverse of the bijection $\exp : R \to (0, \infty)$ (The natural logarithm as the inverse of the exponential function).

Proof

technique · direct

1.1

By [F1] and [L1], $L$ is an inverse of $\exp$ .

F1L1

2.1

Inverse functions are unique, so [F2] and step 1.1 give $L (x) = \log x$ for every $x > 0$ .

step 1.1F2∎

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

OpenStax, Calculus Volume 1, Section 6.7 (standard reference, not scraped)

The integral logarithm LL is the published natural logarithm