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.

Landau's root limit: log x is the limit of 2^n times (x^(1/2^n) minus 1)

Statement

For every $x > 0$ , $\log x = \lim_{n \to \infty} 2^{n} (x^{1 / 2^{n}} - 1) .$

Facts & Assumptions

Given: A positive real $x$ .

[L1]

$x^{r} = \exp (r \log x)$ for real $r$ , and this agrees with rational powers (Real powers for positive bases, with the zero-base positive-exponent convention, The exponential definition of real powers agrees with the existing rational powers).

[L2]

$\exp^{'} (0) = 1$ , so $\lim_{h \to 0} (\exp h - 1) / h = 1$ (The exponential function is smooth and $(\exp)^{'} = \exp$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[L3]

$(1 / 2)^{n} \to 0$ , and limits respect sums, products, and scalar multiples (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , Algebra of limits: sums, scalar multiples, products and quotients).

Proof

technique · direct

1.1

If $x = 1$ , every displayed summand is $0$ , so the limit is $\log 1 = 0$ .

L1algebra

1.2

Suppose $x \neq 1$ and put $h_{n} = (\log x) / 2^{n}$ . Then $h_{n} \to 0$ and $x^{1 / 2^{n}} = \exp (h_{n})$ .

L1L3

2.1

We have $2^{n} (x^{1 / 2^{n}} - 1) = \log x (\exp (h_{n}) - 1) / h_{n}$ .

step 1.2algebra

3.1

The derivative limit in [L2] makes the right-hand side tend to $\log x$ , proving the claim together with step 1.1.

step 1.1step 2.1L2∎

Depends on

Used by

Dependency tree · two levels

52 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)