> [!NOTE] Lemma > For all $u,v>0,$ $\log(uv)=\log(u)+\log(v)$where $\log(x)$ denotes the [[Real Natural Logarithm Function|real natural logarithm]] of $x.$ **Proof**: By [[Real Exponential Function of Sum]], $e^{\log u+\log v} = e^{\log u} e^{\log v} = uv. $Taking the $\log$ of both sides gives the desired result.