CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 6 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

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

The integral logarithm LL is the published natural logarithm