# exponential

## Definition.

For $x\in\mathbb{R}^{+}$ and $n\in\mathbb{Z}$ we define $x^{n}$ in terms of repeated multiplication. To be more precise, we inductively characterize natural number  powers as follows:

 $x^{0}=1,\quad x^{n+1}=x\cdot x^{n},\quad n\in\mathbb{N}.$

The existence of the reciprocal is guaranteed by the assumption  that $\mathbb{R}$ is a field. Thus, for negative exponents, we can define

 $x^{-n}=(x^{-1})^{n},\quad n\in\mathbb{N},$

where $x^{-1}$ is the reciprocal of $x$.

The case of arbitrary exponents is somewhat more complicated. A possible strategy is to define roots, then rational powers, and then extend by continuity. Our approach is different. For $x\in\mathbb{R}^{+}$ and $p\in\mathbb{R}$, we define the set of all reals that one would want to be smaller than $x^{p}$, and then define the latter as the least upper bound of this set. To be more precise, let $x>1$ and define

 $L(x,p)=\{z\in\mathbb{R}^{+}:z^{n}

We then define $x^{p}$ to be the least upper bound of $L(x,p)$. For $x<1$ we define

 $x^{p}=(x^{-1})^{-p}.$

The exponential operation possesses a number of important properties (http://planetmath.org/PropertiesOfTheExponential), some of which characterize it up to uniqueness.