properties of the exponential

The exponential operation possesses the following properties.

• For $x,y\in {ℝ}^{+},p\in ℝ$ we have

  $${(xy)}^p=x^p{y}^p$$

• For $x\in {ℝ}^{+}$ we have

  $$x^0=1,x^1=x,x^{p+q}=x^p{x}^q,{(x^p)}^q=x^{pq}\mathit{\hspace{1em}\hspace{1em}}p,q\in ℝ.$$

• 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 itsarguments. To be more precise, the following function is continuous:

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

Let us also note that the exponential operation is characterized (inthe sense of existence and uniqueness) by the additivity andcontinuity properties. [Author’s note: One can probably get away withsubstantially less, but I haven’t given this enough thought.]