# exponential

## Preamble.

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

$${x}^{p},x\in {\mathbb{R}}^{+},p\in \mathbb{R}.$$ |

The power index $p$ in the above
expression is called the exponent^{}. We take it as proven that $\mathbb{R}$
is a complete^{}, ordered field. No other properties of the real numbers
are invoked.

## 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:

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

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

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

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

$$ |

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 of important properties (http://planetmath.org/PropertiesOfTheExponential), some of which characterize it up to uniqueness.

## Note.

It is also possible to define the exponential operation in
terms of the exponential function^{} and the natural logarithm^{}. Since these concepts require
the context of
differential^{} 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 |