\begin{definition}[Definition:Power (Algebra)/Real Number/Definition 1]
Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $r \in \R$ be a real number.
We define $x^r$ as:
:$x^r := \map \exp {r \ln x}$
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:
:$\forall r \in \Q: \map \exp {r \ln x} = \map \exp {\map \ln {x^r} } = x^r$
{{WIP|The above assertion is Definition:Power (Algebra)/Real Number/Definition 2: write Exponentiation to Real Number is Extension of Exponentiation to Rational Number to prove it, and invoke it as part of the equivalence definition}}
\end{definition}