Logarithm is Strictly Increasing/Corollary

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, $\struct {\R_{> 0}, \le}$ is totally ordered.

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

$\blacksquare$