Bounds of Natural Logarithm

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Theorem

BoundsOfNatLog.png

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:

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

$\blacksquare$


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