\begin{definition}[Definition:Exponential Function/Real/Inverse of Natural Logarithm]
Consider the natural logarithm $\ln x$, which is defined on the open interval $\openint 0 {+\infty}$.
From Logarithm is Strictly Increasing:
:$\ln x$ 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 $\R_{>0}$.
\end{definition}