Logarithm is Strictly Increasing/Corollary

Corollary to Logarithm is Strictly Increasing and Strictly Concave
Let $\ln$ be the natural logarithm.

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

Proof
From Logarithm is Strictly Increasing and Strictly Concave, $\ln$ 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.