Logarithm is Strictly Increasing/Corollary

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Corollary to Logarithm is Strictly Increasing

Let $\ln$ be the natural logarithm.


Then $\ln$ is injective on $\R_{>0}$.


Proof

From Logarithm is Strictly Increasing, $\ln$ is strictly increasing on $\R_{> 0}$.

From Ordering on Real Numbers is Total Ordering, $\left({\R_{> 0}, \le}\right)$ is totally ordered.

From Strictly Monotone Mapping with Totally Ordered Domain is Injective, $\ln$ is injective.

$\blacksquare$