The [[Exponential]] [[Function]] is define as the *unique* [[Real Numbers|real]] [[Function|function]] that maps $0$ to $1$, and has a [[Derivative]] everywhere equal to itself. The exponential is denoted as $e^{x}$ or $\exp(x)$. The defining two properties of the exponent for [[Real Numbers|Real Numbers]] and extended definitions are: $\huge \begin{align} \deriv{ }x \pa{e^{x}} &= e^{x} \\ (e^{a} )(e^{b}) &= e^{{a+b}} \end{align} $ The [[Taylor Series]] for $e^{x}$ is the following: $\huge e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots $ Or more explicitly as this infinite sum: $\huge e^{x } = \sum^{\infty}_{n=0} \frac{x^{n}}{n!} $ This taylor series is what it used to extend this function to more exotic inputs, such as the [[Complex Exponential]] or [[Matrix Exponential|Matrix Exponentials]] ### Original Definition regarding Natural Numbers Exponentiation is a [[Binary Operation]] originally defined between two [[Natural Numbers]] $m$ and $n$ such that the exponential between them, denoted as $m^n$ is equal to the repeated multiplication of $m$, $n$ times. $\huge m^{n} = \prod_{i=1}^{n} m = \underbrace{m \times m \times \cdots \times m}_{n \text{ times}} $ Although only being originally defined on real numbers, this definition is often extended to other domains. This is done typically by ensuring the following core principle: $\huge a^{b} \cdot a^{c} = a^{b+c} $