LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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 is bounded above by its positive argument minus one

Statement

For every real $x \geq 1$ , the natural logarithm of The natural logarithm as the inverse of the exponential function satisfies $0 \leq \log x \leq x - 1 \leq x .$

Facts & Assumptions

Given: $x \geq 1$ and the natural logarithm.

[F1]

The natural logarithm as the inverse of the exponential function: $\log$ is the inverse of the exponential function on $(0, \infty)$ .

[F2]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is continuous and strictly increasing, and $\log 1 = 0$ .

[F3]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log^{'} (t) = 1 / t$ for every $t > 0$ .

[F4]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : a function continuous on a closed interval and differentiable in its interior has a point where its derivative equals its endpoint difference divided by the interval length.

[A1]

$x \geq 1$ .

Proof

Given: The hypotheses in the Statement.

1.1givenA1F1F2F3F4caseschoose

By [F1], $\log$ is defined on $[1, x]$ . If $x = 1$ , [F2] gives $\log x = 0$ . If $x > 1$ , [F2] gives continuity of $\log$ on $[1, x]$ , and [F3] gives differentiability on $(1, x)$ . By [F4], there is $c \in (1, x)$ such that $\log x - \log 1 = \frac{x - 1}{c},$ so $\log x = (x - 1) / c$ .

2.1givenA1F2step 1.1algebracases∎

In the $x = 1$ case the claimed inequalities are equalities except $x - 1 \leq x$ . In the $x > 1$ case, $c > 1$ and $x - 1 > 0$ , so $0 < (x - 1) / c < x - 1 < x$ . Combining with step 1.1 proves all three inequalities.

Depends on

Used by

Dependency tree · two levels

20 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

Jiří Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)