Definition:Exponential Function/Real/Inverse of Natural Logarithm

Definition
Consider the natural logarithm $\ln x$, which is defined on the open interval $\left({0 \,.\,.\, +\infty}\right)$.

From Logarithm is Strictly Increasing and Strictly Concave, $\ln x$ is strictly increasing.

From Inverse of Strictly Monotone Function, the inverse of $\ln x$ always exists.

The inverse of the natural logarithm function is called the exponential function, which is denoted as $\exp$.

Thus for $x \in \R$, we have:
 * $y = \exp x \iff x = \ln y$

The number $\exp x$ is called the exponential of $x$.

The domain of $\exp$ is $\R$, and the codomain of $\exp$ is $\R_{>0}$.