Exponential Function is Well-Defined/Real/Proof 4
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Theorem
Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then $\exp x$ is well-defined.
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.
$\blacksquare$