Definition:Exponential Function

Inverse of the Logarithm
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 $x$.

Exponential in terms of Euler's Number
From the definition of powers for real numbers, we have $z^x = \exp \left({x \ln z}\right)$.

Suppose $z = e$, where $e$ is Euler's number, i.e. $2.71828\ldots$

From that definition of $e$, we have $\ln e = 1$.

Thus $e^x = \exp \left({x \ln e}\right) = \exp x$.

Thus $\exp x$ can be (and frequently is) written and defined as $e^x$.

So the number $e^x$ is also called the exponential of $x$ and the operation of raising $e$ to the power of $x$ is known as the exponential function.

Complex Numbers
The definition still holds when $x \in \C$ is a complex number.