> [!NOTE] Lemma > For all $x>0,$ $\log(x)\leq x-1$where $\log$ denotes the [[Real Natural Logarithm Function|real natural logarithm]]. Moreover $x-1$ is the tangent to $\log x$ at $x=1.$ **Proof**: Let $x>0$ By [[Lower Bound for Real Exponential Function]], for all $t\in \mathbb{R},$ $e^{t}\geq 1+t.$ Taking $t=\log x$ gives $x \geq 1+ \log x \implies \log x \leq x-1.$