Exponential Function is Well-Defined/Real/Proof 4

Proof
This proof assumes the definition of the exponential as the inverse of the logarithm.

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

From Inverse of Strictly Monotone Function, $f$ permits an inverse mapping.

Hence the result, from Inverse Mapping is Unique.