We use + to denote the set of positive real numbers. Our aim is to define the exponentialMathworldPlanetmathPlanetmath, or the generalized power operation,


The power index p in the above expression is called the exponentMathworldPlanetmath. We take it as proven that is a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, ordered field. No other properties of the real numbers are invoked.


For x+ and n we define xn in terms of repeated multiplication. To be more precise, we inductively characterize natural numberMathworldPlanetmath powers as follows:


The existence of the reciprocal is guaranteed by the assumptionPlanetmathPlanetmath that is a field. Thus, for negative exponents, we can define


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+ and p, we define the set of all reals that one would want to be smaller than xp, 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+:zn<xm for all m,n such that m<pn}.

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


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


It is also possible to define the exponential operation in terms of the exponential functionDlmfDlmfMathworldPlanetmath and the natural logarithmMathworldPlanetmathPlanetmathPlanetmath. Since these concepts require the context of differentialMathworldPlanetmath theory, it seems preferable to give a basic definition that relies only on the foundational property of the reals.

Title exponential
Canonical name Exponential
Date of creation 2013-03-22 12:29:59
Last modified on 2013-03-22 12:29:59
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 17
Author rmilson (146)
Entry type Definition
Classification msc 26A03
Synonym exponential operation
Related topic RealNumber
Defines exponent
Defines power