$$\log y\ge1-\frac{1}{y}, \ \forall y>0$$

$\forall x>0$: $$\begin{align*} &\log_bx\le\frac{1}{\log b}(x-1) \\ &\log_bx\ge\frac{1}{\log b}(1-\frac{1}{x}) \end{align*}$$