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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Natural Exponential Function">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 23 17:00:35 1997}
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\section{The Natural Exponential Function\label%
{The Natural Exponential Function}}

Any positive number can be used as the base for an exponential function, but
some bases are used more frequently than others. The bases $2$ and $10$ are
convenient for certain applications. But the most important base is the
number denoted by the letter $e$.

\label{Definition of the number e}The number $\mathbf{e}$ is defined as the
value that $(1+1/n)^{n}$ approaches as $n$ becomes large. (In calculus this
idea is made more precise through the concept of a limit.) The following
table shows the values of the expression $(1+1/n)^{n}$ for increasingly
large values of $n$.

\begin{center}
\begin{tabular}{rl}
$n$ & $(1+\dfrac{1}{n})^{n}$ \\ 
$1$ & \multicolumn{1}{r}{$2.00000$} \\ 
$5$ & \multicolumn{1}{r}{$2.48832$} \\ 
$10$ & \multicolumn{1}{r}{$2.59374$} \\ 
$100$ & \multicolumn{1}{r}{$2.70481$} \\ 
$1000$ & \multicolumn{1}{r}{$2.71692$} \\ 
$10,000$ & \multicolumn{1}{r}{$2.71815$} \\ 
$100,000$ & \multicolumn{1}{r}{$2.71827$} \\ 
$1,000,000$ & \multicolumn{1}{r}{$2.71828$} \\ 
$10,000,000$ & \multicolumn{1}{r}{$2.71828$}
\end{tabular}
\end{center}

The \textbf{natural exponential function} is the exponential function

\[
f(x)=e^{x} 
\]

with base $e$. It is often referred to as \textit{the} exponential function.

\subsection{\protect\vspace{1pt}Compound Interest\label{Compound Interest}}

\textbf{Compound interest} is calculated by the formula

\[
A=P\left( 1+\frac{r}{n}\right) ^{nt} 
\]

where 
\begin{tabular}{l}
$P$ = principal \\ 
$r$ = interest rate \\ 
$n$ = number of times interest is compounded per year \\ 
$t$ = number of years \\ 
$A$ = amount after $t$ years
\end{tabular}

\subsection{\protect\vspace{1pt}Continuously Compounded Interest\label%
{Continuously compounded interest}}

\vspace{1pt}\textbf{Continuously compounded interest} is calculated by the
formula

\[
A=Pe^{rt} 
\]

where 
\begin{tabular}{l}
$P$ = principal \\ 
$r$ = interest rate \\ 
$t$ = number of years \\ 
$A$ = amount after $t$ years
\end{tabular}

\subsection{\protect\vspace{1pt}Exponential Growth\label{Exponential Growth}}

\vspace{1pt}A population that experiences \textbf{exponential growth}
increases according to the formula

\[
n(t)=n_{0}e^{rt} 
\]

where 
\begin{tabular}{l}
$n_{0}$ = initial size of the population \\ 
$r$ = relative rate of growth (expressed as a proportion of the population)
\\ 
$t$ = time \\ 
$n(t)$ = population at time $t$%
\end{tabular}

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