Definition:Exponential Function

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

From Basic Properties of Natural Logarithm, $$\ln x$$ 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 and is written $$\exp$$.

Thus we have $$y = \exp x \iff x = \ln y$$.

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

The definition still holds when $$x \in \mathbb{C}$$ is a complex number.