Definition:Power (Algebra)/Real Number

Definition
Let $x \in \R$ be a real number such that $x > 0$.

Let $r \in \R$ be any real number.

Then we define $x^r$ as:


 * $x^r := \exp \left({r \ln x}\right)$

where $\exp$ denotes the exponential function.

This definition is an extension of the definition for rational $r$.

This follows from Logarithms of Powers and Exponential of Natural Logarithm: it can be seen that:
 * $\forall r \in \Q: \exp \left({r \ln x}\right) = \exp \left({\ln \left({x^r}\right)}\right) = x^r$

Complex Number
This definition can be extended to complex $r$:

Also see

 * Exponentiation to Real Number is Extension of Exponentiation to Rational Number