TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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The exponential definition of real powers agrees with the existing rational powers

Statement

If $a > 0$ and $r \in Q$ , then the real power $a^{r} = \exp (r \log a)$ agrees with the rational power of Rational powers $a^{r}$ of a positive base. For $r > 0$ , both conventions also give $0^{r} = 0$ .

Facts & Assumptions

Given: A positive real $a$ and a rational $r = p / q$ with $q \geq 1$ .

[L1]

Rational powers satisfy $(a^{p / q})^{q} = a^{p}$ , are positive for positive base, and obey the rational power laws (Rational powers $a^{r}$ of a positive base, Laws of rational exponents).

[L2]

For positive reals, $\log (x y) = \log x + \log y$ , $\log (1 / x) = - \log x$ , and $\log$ is injective (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[L3]

The real-power definition is $a^{u} = \exp (u \log a)$ (Real powers for positive bases, with the zero-base positive-exponent convention).

Proof

technique · direct

1.1

For $q \geq 1$ , the product law for $\log$ gives $\log ((a^{p / q})^{q}) = q \log (a^{p / q}) = p \log a = \log (a^{p})$ .

L1L2

2.1

Injectivity of $\log$ gives $\log (a^{p / q}) = (p / q) \log a$ , and exponentiating gives $a^{p / q} = \exp ((p / q) \log a)$ .

step 1.1L2L3

3.1

This is the new real power at exponent $p / q$ ; negative rational exponents follow by reciprocals, and the stated $0^{r}$ convention agrees with Rational powers $a^{r}$ of a positive base for $r > 0$ .

step 2.1L1L3∎

Depends on

Used by

Dependency tree · two levels

23 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)