Bounds of Natural Logarithm

Theorem
Let $\ln y$ be the natural logarithm of $y$ where $y \in \R_{>0}$.

Then $\ln$ satisfies the compound inequality:


 * $\displaystyle 1 - \frac 1 y \le \ln y \le y - 1$

Proof

 * Upper Bound of Natural Logarithm: $\ln y \le y - 1$
 * Lower Bound of Natural Logarithm: $1 - \dfrac 1 y \le \ln y$