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\begin{document}


\section{Exponential Functions and Their Derivatives}

Two of the most important functions that occur in mathematics and its
applications are the exponential function $f(x)=a^{x}$ and its inverse
function, the logarithmic function $g(x)=\log _{a}x$. These functions and
their derivatives are used to describe exponential growth and decay in such
sciences as chemistry, physics, biology, and economics.

\subsubsection{Exponential Functions}

An \textbf{exponential function} is a function of the form 
\[
f(x)=a^{x} 
\]
where $a$ is a positive constant. It is defined in five stages:

\begin{enumerate}
\item[1.]  If $x=n$, a positive integer, then 
\[
a^{n}=\,\stackunder{n\text{ factors}}{\underbrace{a\cdot a\cdot \cdots \cdot
a}}
\]

\item[2.]  If $x=0,$ then $a^{0}=1$.

\item[3.]  If $x=-n$, $n$ a positive integer, then 
\[
a^{-n}=\frac{1}{a^{n}}
\]

\item[4.]  If $x$ is a rational number, $x=\dfrac{p}{q}$, where $p$ and $q$
are integers and $q>0$, then 
\[
a^{x}=a^{\frac{p}{q}}=\sqrt[q\,]{a^{p}}
\]

\item[5.]  If $x$ is an irrational number, $a^{x}$ is defined as a limit 
\[
a^{x}=\lim\limits_{r\stackrel{}{\rightarrow }x}\,\,a^{r}\qquad r\text{
rational }
\]
\end{enumerate}

It can be shown that this definition uniquely specifies $a^{x}$ and makes
the function $f(x)=a^{x}$ continuous.

\begin{example}
The number $\sqrt{2}$ is an irrational number that has a nonrepeating
decimal representation: 
\[
\sqrt{2}=1.414213562373095...
\]
This means that $\sqrt{2}$ is the limit of the sequence of rational numbers 
\[
1\text{,}\quad 1.4\text{,}\quad 1.41\text{,}\quad 1.414\text{,}\quad 1.4142%
\text{,}\quad 1.41421\text{,}\quad 1.414213\text{,}\quad ...
\]
Thus $a^{\sqrt{2}}$ is the limit of the sequence of rational powers of $a$ 
\[
a^{1}\text{,}\quad a^{1.4}\text{,}\quad a^{1.41}\text{,}\quad a^{1.414}\text{%
,}\quad a^{1.4142}\text{,}\quad a^{1.41421}\text{,}\quad a^{1.414213}\text{,}%
\quad ...
\]
\end{example}

\subsubsection{Properties of Exponential Functions\label%
{Exponential functions}}

\begin{theorem}
If $a>0$ and $a\neq 1$, then $f(x)=a^{x}$ is a continuous function with
domain $R$ and range $(0,\infty )$. In particular, $a^{x}>0$ for all $x$. If 
$0<a<1$, then $f(x)=a^{x}$ is a decreasing function; if $a>1$, then $f$ is
an increasing function. If $a$, $b>0$ and $x$, $y\in R$, then 
\[
a^{x+y}=a^{x}a^{y}\qquad \quad a^{x-y}=\frac{a^{x}}{a^{y}}\qquad \quad
(a^{x})^{y}=a^{xy}\qquad \quad (ab)^{x}=a^{x}b^{x}
\]
\end{theorem}

\subsubsection{Limits of Exponential Functions\label%
{Limits of exponential functions}}

\begin{theorem}
If $a>1$, then\qquad $\lim\limits_{x\rightarrow \infty }\,\,a^{x}=\infty $%
\qquad and\qquad $\lim\limits_{x\rightarrow \,-\,\infty }\,\,a^{x}=0$

If $0<a<1$, then\qquad $\lim\limits_{x\rightarrow \infty }\,\,a^{x}=0$\qquad
and\qquad $\lim\limits_{x\rightarrow \,-\,\infty }\,\,a^{x}=\infty $
\end{theorem}

\subsubsection{\QTR{BOXHEAD}{Properties of the Exponential Function Base }$e$%
}

The exponential function $f(x)=e^{x}$ is an increasing continuous function
with domain $R$ and range $(0,\infty )$. Thus $e^{x}>0$ for all $x$. Also 
\[
\lim\limits_{x\rightarrow \,-\,\infty }\,\,e^{x}=0\qquad \lim_{x\rightarrow
\infty }\,\,e^{x}=\infty 
\]
So the $x$-axis is a horizontal asymptote of $f(x)=e^{x}$.

\subsection{Related Reference Topics}

{\small \hyperref{Exponential functions of one real variable, definition}{}{%
}{Al05_01.tex#Exponential Functions}}

{\small \hyperref{Exponential growth, algebra}{}{}{%
Al05_02.tex#Exponential Growth}}

{\small \hyperref{Natural exponential function}{}{}{%
Al05_02.tex#The Natural Exponential Function}}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Exponential functions}{}{}{../../Help/DM3-5.tex#Exponents}}

\noindent \hrulefill

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