TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

Change of base and inversion of the positive-base real exponential

Statement

If $b > 0$ , $b \neq 1$ , and $x > 0$ , then $\log_{b} x = \frac{\log x}{\log b}, b^{\log_{b} x} = x, \log_{b} (b^{u}) = u (u \in R) .$ In particular, $\log_{b} x = \log_{c} x / \log_{c} b$ for every second base $c > 0$ , $c \neq 1$ .

Facts & Assumptions

Given: $b, c > 0$ with $b, c \neq 1$ , $x > 0$ , and $u \in R$ .

[L1]

$\log_{b} x = \log x / \log b$ , with $\log b \neq 0$ (The logarithm to a positive base other than one).

[L2]

$b^{v} = \exp (v \log b)$ and the real-power laws hold for positive bases (Real powers for positive bases, with the zero-base positive-exponent convention, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents).

[L3]

$\log (\exp v) = v$ and $\exp (\log x) = x$ for $x > 0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

Proof

technique · direct

1.1

By [L1] and [L2], $b^{\log_{b} x} = \exp ((\log x / \log b) \log b) = \exp (\log x) = x$ .

L1L2L3

1.2

Likewise $\log_{b} (b^{u}) = \log (\exp (u \log b)) / \log b = u$ .

L1L2L3

2.1

Dividing $\log x$ first by $\log c$ and then by $\log b / \log c$ gives $\log_{b} x = \log_{c} x / \log_{c} b$ .

L1algebra∎

Depends on

Used by

Dependency tree · two levels

10 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

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)