Definition:Exponential Function/Real/Extension of Rational Exponential

Definition
Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function. From the definition of powers for real numbers, we have for all $a \in \R_{>0}, x \in \R$:
 * $a^x = \exp \left({x \ln a}\right)$

Suppose $a = e$, 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.