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.