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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:13:15 1997}
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\begin{document}


\section{Exponents\label{Exponents} and Logarithms}

In \textsl{Scientific Notebook} you can work with 
\index{Exponential function@Exponential function} exponential and
logarithmic functions 
\index{Logarithmic functions@Logarithmic functions}\label%
{Logarithmic functions}in their natural forms: $e^{x}$, $\exp x$, $\log
_{5}x $, $\ln x$, and so forth. These functions are inverses of one another,
as exemplified by the identities $e^{\ln x}=x$ and $\ln e^{x}=x$.

\strut

\subsection{\ Exponents\label{Exponential function} and Exponential Functions
}

Exponential functions are used in modeling many real life situations. The
laws of exponents are an important feature of these functions.

\subsubsection{Laws of Exponents}

You can demonstrate the laws of exponents with either \textsf{Simplify} or 
\index{Simplify@Simplify}\textsf{Combine + Powers}. You can also use \textsf{%
Combine + Exponential} for 
\index{Combine@Combine!exponential@exponential}\label{Combine, exponentials} 
\index{Combine@Combine!powers@powers}\label{Combine, powers}the first of
these equations. These laws work for real or complex exponents and for other
expressions as well.

\begin{quote}
$\blacktriangleright $ \textsf{Simplify} (or \textsf{Combine + Powers})
\end{quote}

\begin{quotation}
$2^{x}2^{y}=\,2^{x+y}$\qquad \qquad

$\left( 3^{x}\right) ^{y}=3^{xy}$\qquad \qquad

$%
\dfrac{a^{x}}{a^{y}}=a^{x-y}$
\end{quotation}

\textsl{Scientific Notebook} normally returns exponential expressions in the
form $\,e^{f(x)}$ rather than $\exp f(x)$. However, when the exponent $f(x)$
is sufficiently complicated, the linear form $\exp f(x)$ is returned. The
following example illustrates these two behaviors.

\begin{quote}
$\blacktriangleright $ \textsf{Simplify} (or \textsf{Combine + Powers})
\end{quote}

\begin{quotation}
$\left( e^{a+b}\right) ^{3c}=\,e^{3\left( a+b\right) c}$

$e^{x^{2}-3xy}e^{2x+5}=\,\exp \left( x^{2}-3xy+2x+5\right) $
\end{quotation}

\strut

\subsubsection{Evaluating Exponential Functions}

Use \textsf{Evaluate} or \textsf{Evaluate Numerically} for numerical
approximations. Note that the use of floating-point notation triggers a
numeral evaluation. Change the \textsf{Settings} dialog box for the number
of digits you want. In these examples, \textsf{Digits Used in Display} is
set to $8$.

\begin{center}
{\ 
\begin{tabular}{lll}
\textbf{Expression} & \textbf{Evaluate} & \textbf{Evaluate Numerically} \\ 
\hline
$e^{2}$ & $e^{2}$ & $\,7.3890561$ \\ 
$e^{0.0025}$ & $\,1.0025031$ & $\,1.0025031$ \\ 
$5^{4}$ & $625$ & $625.0$ \\ 
$2^{\sqrt{5}}$ & $\,2^{\sqrt{5}}$ & $\,4.7111131$%
\end{tabular}
}
\end{center}

\strut

\subsection{\ Logarithms and Logarithmic Functions \label{Logarithms}}

The function $\ln x$ is interpreted as the natural logarithm base $e$.
Logarithms to other bases are entered with a subscript on the function $\log 
$. Thus, evaluation gives $\log _{5}25=\,2$ and $\log _{10}10^{3}=\,3$. The
symbol $\log x$ is interpreted as the natural logarithm (base $e$) unless
you make a change in the \textsf{Maple Settings} dialog.

\begin{quote}
$\blacktriangleright $ To change the default for the function name $\log $
from base $e$ to base $10$
\end{quote}

\begin{enumerate}
\item[1.]  From the menu \textsf{Maple}, choose\textsf{\ Settings.}

\item[2.]  Choose \textsf{Definition Options}.

\item[3.]  Under \textsf{Base for Log Function}, check \textsf{Change from e
to 10}.

\item[4.]  Choose \textsf{OK}.
\end{enumerate}

All logarithmic functions are converted symbolically to natural logarithms
by evaluation as follows.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\log _{10}x=\,\dfrac{\ln x}{\ln 2+\ln 5}\vspace{6pt}$

$\log _{2}x=\,\dfrac{\ln x}{\ln 2}\vspace{6pt}$

$\log _{\frac{1}{2}}x=\,-\dfrac{\ln x}{\ln 2}\vspace{12pt}$
\end{quotation}

\strut

Thus, the behavior of the system for natural logarithms is of particular
interest.\vspace{3pt}

\subsubsection{Properties of Logarithms\label{Combine, logarithms}}

You can demonstrate properties of logarithms with \textsf{Simplify} and 
\textsf{Combine}. 
\index{Combine@Combine!logarithms@logarithms}

\begin{quote}
$\blacktriangleright $ \textsf{Simplify}
\end{quote}

\begin{quotation}
$\ln x^{y}=\,y\ln x%
\vspace{12pt}$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Combine} \textsf{+ Logs}
\end{quote}

\begin{quotation}
$\ln x+\ln y=\ln xy\vspace{3pt}$

$\ln a-\ln b=\ln \frac{a}{b}\vspace{12pt}$
\end{quotation}

\subsubsection{Evaluating Logarithms}

Use \textsf{Evaluate} or \textsf{Evaluate Numerically} for numerical
approximations. Note that \textsf{Evaluate} converts to natural logarithms
symbolically, and use of floating-point notation triggers a numerical
evaluation. Change the \textsf{Settings} dialog box for the degree of
accuracy you need. In these examples, \textsf{Digits Used in Display} is set
to $5$.

\begin{center}
{\ 
\begin{tabular}{lll}
\textbf{Expression} & \textbf{Evaluate} & \textbf{Evaluate Numerically} \\ 
\hline
$\ln 2$ & $\ln 2$ & $\,.69315$ \\ 
$\log _{10}5$ & $\,\frac{\ln 5}{\ln 2+\ln 5}$ & $\,\,.69897$ \\ 
$\ln 0.0025$ & $\,-5.9915$ & $-5.9915$ \\ 
$\ln 1.0025$ & $2.4969\times 10^{-3}$ & $2.4969\times 10^{-3}$%
\end{tabular}
}
\end{center}

\strut

\subsection{\ Solving Exponential \label{Solve, exponential equations}and
Logarithmic Equations}

For symbolic solutions to exponential or logarithmic equations, from the 
\textsf{Solve} submenu, choose \textsf{Exact}. Enter \textsf{Variable(s) to
Solve for} if requested. For numerical solutions, you can either enter a
coefficient in decimal notation and choose \textsf{Solve + Exact}, or apply 
\textsf{Evaluate Numerically\ }to the symbolic solutions. In the case of a
single variable, you can choose \textsf{Solve + Numeric}.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$3^{x}=8$, Solution is : $\left\{ x=\frac{\ln 8}{\ln 3}\right\} $

$e^{x}=\dfrac{y+1}{y-1}$, Solution is : $\left\{ x=\ln \left( \frac{y+1}{y-1}%
\right) \right\} $ {\ (Solve for }$x${)}

$P=Qe^{kt}$, Solution is : $\left\{ k=\frac{\ln \frac{P}{Q}}{t}\right\} $
(Solve for $k$)

$\log \left( 3x+y\right) =8$, Solution is : $\left\{ x=\frac{1}{3}e^{8}-%
\frac{1}{3}y\right\} $ (Solve for $x$)

$\log _{5}\left( 4x^{2}-3y\right) =5^{\frac{5}{\ln 5}}$, Solution is : $%
\left\{ x=-\frac{1}{2}\sqrt{\left( e^{5^{\frac{5}{\ln 5}}\ln 5}+3y\right) }%
\right\} ,$

$\allowbreak \qquad \allowbreak \qquad \allowbreak \qquad \allowbreak \qquad
\left\{ x=\frac{1}{2}\sqrt{\left( e^{5^{\frac{5}{\ln 5}}\ln 5}+3y\right) }%
\right\} $ (Solve for $x$)
\end{quotation}

{\ }

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Solving systems of equations}{}{}{DM3-3.tex#Systems of
equations}

\item  \hyperref{2D Plots of Expressions}{}{}{DM6-4.tex#Plot expression}

\item  \hyperref{Exercises and Solutions}{}{}{DM3-6.tex#Exercises}
\end{itemize}

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DMIndex.tex#Exponential functions}, \hyperref{Logarithmic functions}{}{}{%
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\end{document}
