PlanetMath: properties of the exponential

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properties of the exponential

(Theorem)

The exponential operation possesses the following properties.

Homogeneity. For $x,y\in\mathbb{R}^+, p\in \mathbb{R}$ we have
$\displaystyle (xy)^p = x^p y^p$
Exponent additivity. For $x\in\mathbb{R}^+$ we have
$\displaystyle x^0=1,\quad x^1 = x,\quad x^{p+q} = x^p x^q,\quad (x^p)^q=x^{pq}\qquad p,q\in\mathbb{R}.$
Monotonicity. For $x,y\in\mathbb{R}^+$ with and $p\in \mathbb{R}^+$ we have
$\displaystyle x^p < y^p,\qquad x^{-p} > y^{-p}.$
Continuity. The exponential operation is continuous with respect to its arguments. To be more precise, the following function is continuous:
$\displaystyle P:\mathbb{R}^+\times\mathbb{R}\rightarrow \mathbb{R},\qquad P(x,y)=x^y.$

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.]

"properties of the exponential" is owned by rmilson.

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Attachments:: proof of properties of the exponential (Proof) by rspuzio

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Cross-references: additivity, function, arguments, continuous, properties, exponential operation
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This is version 12 of properties of the exponential, born on 2002-02-27, modified 2004-09-29.
Object id is 2731, canonical name is PropertiesOfTheExponential.
Accessed 4993 times total.

Classification:

AMS MSC:

26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line)

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