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 Logarithm is Strictly Increasing and 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 and is written $\exp$.

Thus we have:
 * $y = \exp x \iff x = \ln y$

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

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.

In Terms of a Limit
The exponential function can also be defined as the following limit:

$e^x := \displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$.

Using this definition, the natural logarithm is then defined as the function inverse of the exponential function, which can be shown to exist.

In Terms of a Differential Equation
The exponential function can be defined as the unique solution $y = f(x)$ to the first order ODE

$\dfrac{\mathrm d y}{\mathrm d x} = y$

satisfying the condition $f(0) = 1$

Informally (and a bit imprecisely), this states that the exponential function is defined as the function that is its own derivative.

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

Linguistic Note
The word exponential derives ultimately from the (now archaic) verb to expone, which means to set forth, in the sense of to expound, or explain.

This itself comes from the Latin expono, meaning I expose, or I exhibit.

The word exponent (from which exponential is formed) therefore means a person (or statement) that explains something.