TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 115 results over 32 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

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)