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