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.

$\blacksquare$