properties of the exponential
The exponential operation possesses the following properties.

•
For $x,y\in {\mathbb{R}}^{+},p\in \mathbb{R}$ we have
$${(xy)}^{p}={x}^{p}{y}^{p}$$ 
•
For $x\in {\mathbb{R}}^{+}$ we have
$${x}^{0}=1,{x}^{1}=x,{x}^{p+q}={x}^{p}{x}^{q},{({x}^{p})}^{q}={x}^{pq}\mathit{\hspace{1em}\hspace{1em}}p,q\in \mathbb{R}.$$ 
•
Monotonicity. (http://planetmath.org/TotalOrder) For $x\mathrm{,}y\mathrm{\in}{\mathrm{R}}^{\mathrm{+}}$ with $$ and $p\mathrm{\in}{\mathrm{R}}^{\mathrm{+}}$ we have
$$ 
•
Continuity. The exponential operation is continuous^{} with respect to its arguments. To be more precise, the following function^{} is continuous:
$$P:{\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R},P(x,y)={x}^{y}.$$
Let us also note that the exponential operation is characterized (in the sense of existence and uniqueness) by the additivity and continuity properties. [Author’s note: One can probably get away with substantially less, but I haven’t given this enough thought.]
Title  properties of the exponential^{} 

Canonical name  PropertiesOfTheExponential 
Date of creation  20130322 12:30:02 
Last modified on  20130322 12:30:02 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  15 
Author  rmilson (146) 
Entry type  Theorem 
Classification  msc 26A03 