> [!Definition] Definition > Let $x>0$ and $p\in\mathbb{R}.$ We define the $p$th power of $x$ as follows $x^{p}=\exp(p\log x)$where $\exp$ and $\log$ denote the [[Real Exponential Function|real exponential function]] and [[Real Natural Logarithm Function|real natural logarithm function]] respectively. # Properties By [[Real Power Extends Integer Power of Real Number]], for all $n\in \mathbb{N}^{+},$ $x^{n}=\underbrace{x\cdot x\cdots x}_{\text{n } times}.$ If $n$ is a positive integer then [[Integer Power of Real Number|power]] $x^n$ as defined here is indeed the product $x . x \ldots x$ of $n$ copies of $x$. >*Proof*. $x^{p+q}=x^p x^q$ for all $x>0$ and $p, q \in \mathbb{R}$. >*Proof* $\log \left(x^p\right)=p \log x$ for $x>0$ and $p \in \mathbb{R}$. >*Proof* $x^{p q}=\left(x^p\right)^q$ for all $x>0$ and $p, q \in \mathbb{R}$. >*Proof.* The $p$th power of [[Euler's Number|Euler's number]] is given by the [[Real Exponential Function|exponential function]] on $p$ i.e. $\exp (p)=e^p$ for all $p \in \mathbb{R}$. > *Proof*. By first definition of power, $e^{p}=\exp (p\log(e))$Note that $e=\exp(1)$ so $\log(e)=\log(\exp(1))=1$ so indeed $e^{p}=\exp(p)$.