Image of Real Natural Logarithm

Theorem

Let $\ln$ be the natural logarithm function on the real numbers.

Then the image of $\ln$ is the set of real numbers:

$\Img \ln = \R$

Proof

By definition of natural logarithm:

$\ln^{-1} = \exp$

From Exponential Tends to Zero and Infinity:

$\Dom \exp = \openint {-\infty} {+\infty}$

$\Img \exp = \openint 0 {+\infty}$

From Exponential is Strictly Increasing:

$\exp$ is strictly increasing.

From Strictly Monotone Real Function is Bijective, $\exp: \R \to \R_{>0}$ is a bijection.

Thus:

$\Img \ln = \Dom \exp$

and so $\Img \ln = \R$.

$\blacksquare$