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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Logarithmic Functions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 13:15:04 1997}
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\section{Logarithmic Functions\label{Logarithmic Functions}}

\vspace{1pt}

\subsection{\protect\vspace{1pt}Definition of the Logarithmic Function\label%
{Definition of the logarithmic function with base a}}

Let $a$ be a positive number with $a\neq 1$. The \textbf{logarithmic
function with base} $a$, denoted by $\log $, is defined by

\[
\log _{a}x=y\Longleftrightarrow a^{y}=x 
\]

In words, this says that

\begin{center}
$\log _{a}x$ is the exponent to which the base $a$ must be raised to give $x$%
.
\end{center}

\vspace{1pt}

\subsection{Properties of Logarithms\label{Properties of Logarithms}}

\begin{tabular}{cc}
\textbf{Property} & \textbf{Reason} \\ 
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} \\ 
\multicolumn{1}{l}{$\log _{a}1=0$} & \multicolumn{1}{l}{We must raise $a$ to
the power $0$ to get $1$} \\ 
\multicolumn{1}{l}{$\log _{a}a=1$} & \multicolumn{1}{l}{We must raise $a$ to
the power $1$ to get $a$.} \\ 
\multicolumn{1}{l}{$\log _{a}a^{x}=x$} & \multicolumn{1}{l}{We must raise $a$
to the power $x$ to get $a^{x}$.} \\ 
\multicolumn{1}{l}{$a^{\log _{a}x}=x$} & \multicolumn{1}{l}{$\log _{a}x$ is
the power to which $a$ must be raised to get $x$.}
\end{tabular}

\subsection{Common Logarithm\label{Common Logarithm}}

The logarithm with base $10$ is called the common logarithm and is denoted
by omitting the base:

\[
\log x=\log _{10}x 
\]

\subsection{Natural Logarithm\label{Natural Logarithm}}

The logarithm with base $e$ is called the natural logarithm and is denoted
by $\ln $:

\[
\ln x=\log _{e}x 
\]

\subsection{Properties of Natural Logarithms\label%
{Properties of Natural Logarithms}}

\begin{tabular}{cc}
\textbf{Property} & \textbf{Reason} \\ 
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} \\ 
\multicolumn{1}{l}{$\ln 1=0$} & \multicolumn{1}{l}{We must raise $e$ to the
power $0$ to get $1$} \\ 
\multicolumn{1}{l}{$\ln e=1$} & \multicolumn{1}{l}{We must raise $e$ to the
power $1$ to get $e$.} \\ 
\multicolumn{1}{l}{$\ln e^{x}=x$} & \multicolumn{1}{l}{We must raise $e$ to
the power $x$ to get $e^{x}$.} \\ 
\multicolumn{1}{l}{$e^{\ln x}=x$} & \multicolumn{1}{l}{$\ln x$ is the power
to which $e$ must be raised to get $x$.}
\end{tabular}

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