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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Logarithmic Functions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 17:48:33 1997}
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\section{Logarithmic Functions\label{Logarithmic functions}}

If $a>0$ and $a\neq 1$, the exponential function $f(x)=a^{x}$ is either
increasing or decreasing and so it is one-to-one. It therefore has an
inverse function $f^{-1}$, which is called the \textbf{logarithmic function
with base }$a$ and is denoted by $\log _{a}$. Using the definition of an
inverse function given by 
\[
f^{-1}(x)=y\quad \Leftrightarrow \quad f(y)=x 
\]
then 
\[
\log _{a}\,x=y\quad \Leftrightarrow \quad a^{y}=x 
\]

\subsubsection{Properties of Logarithmic Functions\label%
{Properties of logarithmic functions}}

\begin{theorem}
If $a>1$, the function $f(x)=\log _{a}\,x$ is a one-to-one, continuous,
increasing function with domain $(0,\infty )$ and range $R$. If $x$, $y>0$,
then
\end{theorem}

\begin{enumerate}
\item[1.]  $\log _{a}(xy)=\log _{a}\,x+\log _{a}y$

\item[2.]  $\log _{a}\,\left( \dfrac{x}{y}\right) =\log _{a}\,x-\log _{a}y$

\item[3.]  $\log _{a}\,(x^{y})=y\log _{a}\,x$

\item  If $a>1$, then 
\[
\lim\limits_{x\rightarrow \infty }\,\,\log _{a}\,x=\infty \qquad \text{%
and\qquad }\lim\limits_{x\rightarrow 0^{+}}\,\,\log _{a}\,x=-\infty 
\]
\end{enumerate}

For any positive number $a$ $(a\neq 1)$, we have 
\[
\log _{a}\,x=\dfrac{\ln x}{\ln a} 
\]
The logarithmic function $\log _{a}$ is continuous since it is the inverse
of a continuous function, namely the exponential function. Its graph is the
reflection of the graph of $y=a^{x}$ about the line $y=x$.$\ $Following are
graphs of logarithmic functions for the bases $e$, $10$, and $-\frac{1}{2}$.%
\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\log _{10}x$};linecolor "blue";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\ln x$};linecolor
"red";linestyle 1;linethickness 1;pointstyle "point";function \TEXUX{$\log
_{.5}x$};linecolor "green";linestyle 1;linethickness 1;pointstyle
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4;constrained TRUE;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};}}The graphs are increasing for $a>1$ and decreasing for $a<1$.
The fact that $y=a^{x}$ is a very rapidly increasing or decreasing function
for $x>0$ is reflected in the fact that $y=\log _{a}x$ is a very slowly
increasing or decreasing function for $x>1$.

\subsubsection{Limits At Infinity\label{Limits at infinity}}

For $f$ a function defined on some interval $(a,\infty )$, 
\[
\lim\limits_{x\rightarrow \infty }\,\,f(x)=L 
\]
means that the values of $f(x)$ can be made arbitrarily close to $L$ by
taking $x$ sufficiently large. For $f$ a function defined on some interval $%
(-\infty ,a)$. Then 
\[
\lim\limits_{x\rightarrow -\infty }\,\,f(x)=L 
\]
means that the values of $f(x)$ can be made arbitrarily close to $L$ by
taking $x$ sufficiently large negative.

\begin{enumerate}
\item  Let $f$ be a function defined on some interval $(a,\infty )$. Then 
\[
\lim\limits_{x\stackrel{}{\rightarrow }\infty }\,\,f(x)=L
\]
means that for every $\varepsilon >0$ there is a corresponding number $N$
such that 
\[
\left| f(x)-L\right| <\varepsilon \qquad \text{whenever}\qquad x>N
\]

\item  Let $f$ be a function defined on some interval $(a,\infty )$. Then 
\[
\lim\limits_{x\rightarrow \infty }\,\,f(x)=\infty 
\]
means that for every positive number $M$ there is a corresponding number $N>0
$ such that 
\[
f(x)>M\text{\qquad whenever\qquad }x>N
\]

\item  Let $f$ be a function defined on some interval $(a,\infty )$. Then 
\[
\lim\limits_{x\rightarrow \infty }\,\,f(x)=-\infty \emph{\ }
\]
means that for every positive number $M$ there is a corresponding number $N>0
$ such that 
\[
f(x)<-M\text{\qquad whenever\qquad }x>N
\]
\end{enumerate}

\noindent Analogous definitions hold for $f$ a function defined on some
interval $\left( -\infty ,a\right) $.

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Definition and notation for limits}{}{}{%
../../Help/DM7-2.tex#Notation for limits}}

{\small \hyperref{Infinite limits}{}{}{../../Help/DM7-2.tex#Infinite limits}}

{\small \hyperref{Inverse function}{}{}{../../Help/DM5-2.tex#Inverse function%
}}

{\small \hyperref{Limit}{}{}{../../Help/DM7-2.tex#Limit}}

{\small \hyperref{Limits at infinity}{}{}{../../Help/DM7-2.tex#Limits at
infinity}}

{\small \hyperref{Log and log log plots}{}{}{../../Help/DM6-4.tex#Log scaling%
}}

{\small \hyperref{Logarithmic functions }{}{}{../../Help/DM3-5.tex#Logarithms%
}}

{\small \hyperref{One-sided limits }{}{}{../../Help/DM7-2.tex#One-sided
limits}}

\noindent \hrulefill \qquad 

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"0.9959";cropright "0.9937";cropbottom "0";filename
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\noindent \hrulefill

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