Surjective Restriction of Real Exponential Function

Theorem

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

$\map \exp x = e^x$

Then the restriction of the codomain of $\exp$ to the strictly positive real numbers:

$\exp: \R \to \R_{>0}$

Hence:

$\exp: \R \to \R_{>0}$

is a bijection.

Proof

Let $y \in \R_{> 0}$.

Then $\exists x \in \R: x = \map \ln y$

That is:

$\exp x = y$

and so $\exp: \R \to \R_{>0}$ is a surjection.

Hence the result.

$\blacksquare$