Real power of a positive number

For $x>0$ and $a\in\mathbb R$ , the real power is

x^a=\exp(a\log x).

This agrees with integer powers and satisfies $x^{a+b}=x^a x^b$ , $(xy)^a=x^ay^a$ for positive $x,y$ , and

\frac{d}{dx}x^a=ax^{a-1}.

The definition uses the real logarithm; negative bases do not admit this definition for arbitrary real exponents.

Endpoints and parameters

For $a>0$ , setting $0^a=0$ gives a continuous extension to zero. Smoothness at zero depends on $a$ . Differentiating the exponent instead gives $\partial_a x^a=x^a\log x$ , explaining logarithmic factors when a power-law exponent varies.

References

John K. Hunter, Introduction to Analysis, Chapter 10: Power Series.