PropertiesOfTheExponential 2002-02-27 09:59:07 2004-09-29 19:47:44 Theorem exponential \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\znums}{\mathbb{Z}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} The exponential operation possesses the following properties. \begin{itemize} \item \PMlinkescapetext{{\bf Homogeneity.}} For $x,y\in\reals^+, p\in \reals$ we have $$(xy)^p = x^p y^p$$ \item \PMlinkescapetext{{\bf Exponent additivity.}} For $x\in\reals^+$ we have $$x^0=1,\quad x^1 = x,\quad x^{p+q} = x^p x^q,\quad (x^p)^q=x^{pq}\qquad p,q\in\reals.$$ \item \PMlinkname{\bf Monotonicity.}{TotalOrder} For $x,y\in\reals^+$ with $x<y$ and $p\in \reals^+$ we have $$x^p < y^p,\qquad x^{-p} > y^{-p}.$$ \item {\bf Continuity.} The exponential operation is continuous with respect to its arguments. To be more precise, the following function is continuous: $$P:\reals^+\times\reals\rightarrow \reals,\qquad P(x,y)=x^y.$$ \end{itemize} Let us also note that the exponential operation is characterized (in the sense of existence and uniqueness) by the {\em additivity} and {\em continuity} properties. [{\bf Author's note}: One can probably get away with substantially less, but I haven't given this enough thought.]