properties of the exponential

The exponential operation possesses the following properties.

  • For x,y+,p we have

  • For x+ we have

    x0=1,x1=x,xp+q=xpxq,(xp)q=xpq  p,q.
  • Monotonicity. ( For x,yR+ with x<y and pR+ we have

  • Continuity. The exponential operation is continuousMathworldPlanetmath with respect to its arguments. To be more precise, the following functionMathworldPlanetmath is continuous:


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 exponentialMathworldPlanetmath
Canonical name PropertiesOfTheExponential
Date of creation 2013-03-22 12:30:02
Last modified on 2013-03-22 12:30:02
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 15
Author rmilson (146)
Entry type Theorem
Classification msc 26A03