> [!NOTE] Lemma > For all $x>0,$ $p,q\in \mathbb{R},$ $x^{pq}=(x^{p})^{q}$where $x^{p}$ denotes the $p$th [[Real Power of Real Number|power]] of $x.$ **Proof**: By definition of $x^{p}$ and [[Real Natural Logarithm Function|log as the inverse of exp]], $(x^{p})^{q}=\exp(q \log(\exp(p\log(x))))=\exp(qp\log(x))=x^{pq}.$