Exponential on Real Numbers is Injection

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Theorem

Let $\exp: \R \to \R$ be the exponential function:

$\map \exp x = e^x$


Then $\exp$ is an injection.

Proof

From Exponential is Strictly Increasing:

$\exp$ is strictly increasing on $\R$.

From Strictly Monotone Mapping with Totally Ordered Domain is Injective:

$\exp$ is an injection.

$\blacksquare$


Sources