> [!NOTE] Lemma > Let $x>0$ and $n\in \mathbb{N}^{+}.$ Then the $n$th [[Real Power of Real Number|power]] of $x$ satisfies $x^{n}=\underbrace{x\cdot x\cdots x}_{\text{n } times}$that is, the definition is consistent with that in [[Integer Power of Real Number|integer power of real number]]. **Proof**: By [[Real Natural Logarithm of Real Power of Real Number]], $\exp(n\log x)=\exp(\log x^{n})=x^{n}$