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:
 * $\operatorname{Im} \left({\ln}\right) = \R$

Proof
By definition of natural logarithm:


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

From Exponential Tends to Zero and Infinity:
 * $\operatorname{Dom} \left({\exp}\right) = \left({-\infty \,.\,.\, +\infty}\right)$
 * $\operatorname{Im} \left({\exp}\right) = \left({0 \,.\,.\, +\infty}\right)$

From Exponential is Strictly Increasing and Convex, $\exp$ is strictly increasing.

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

Thus:
 * $\operatorname{Im} \left({\ln}\right) = \operatorname{Dom} \left({\exp}\right)$

and so $\operatorname{Im} \left({\ln}\right) = \R$.