# 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

 $x^{0}=1,\quad x^{1}=x,\quad x^{p+q}=x^{p}x^{q},\quad(x^{p})^{q}=x^{pq}\qquad p% ,q\in\mathbb{R}.$
• Monotonicity. (http://planetmath.org/TotalOrder) For $x,y\in\mathbb{R}^{+}$ with $x and $p\in\mathbb{R}^{+}$ we have

 $x^{p}y^{-p}.$
• Continuity. The exponential operation is continuous with respect to its arguments. To be more precise, the following function is continuous:

 $P:\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{R},\qquad P(x,y)=x^{y}.$

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 PropertiesOfTheExponential 2013-03-22 12:30:02 2013-03-22 12:30:02 rmilson (146) rmilson (146) 15 rmilson (146) Theorem msc 26A03