Bounds of Natural Logarithm
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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:
- $1 - \dfrac 1 x \le \ln x$
$\blacksquare$