exponential

Preamble.

We use ${ℝ}^{+}\subset ℝ$ to denote the set ofpositive real numbers. Our aim is to define the exponential, orthe generalized power operation,

$$x^p,x\in {ℝ}^{+},p\in ℝ.$$

The power index $p$ in the aboveexpression is called the exponent. We take it as proven that $ℝ$is a complete, ordered field. No other properties of the real numbersare invoked.

Definition.

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

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

The existence of thereciprocal is guaranteed by the assumption that $ℝ$ is a field.Thus, for negative exponents, we can define

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

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

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

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

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

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

Note.

It is also possible to define the exponential operation interms of the exponential function and the natural logarithm. Since these concepts requirethe context ofdifferential theory, it seems preferable to give a basic definitionthat relies only on the foundational property of the reals.