Bounds of Natural Logarithm

Theorem


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

Then $\ln$ satisfies the compound inequality:


 * $1 - \dfrac 1 x \le \ln x \le x - 1$

Proof
From Upper Bound of Natural Logarithm:
 * $\ln x \le x - 1$

From Lower Bound of Natural Logarithm:
 * $1 - \dfrac 1 x \le \ln x$

Also see

 * Bounds for Complex Logarithm