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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Logarithm Defined as an Integral">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 22:30:23 1997}
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\section{The Logarithm Defined as an Integral\label%
{natural logarithmic function}}

\subsubsection{Natural Logarithmic Function}

The \textbf{natural logarithmic function} is the function defined by 
\[
\ln x=\int_{1}^{x}\,\,\frac{1}{t}\,\,dt\qquad x>0 
\]

\subsubsection{\QTR{BOXHEAD}{Laws of Logarithms\label{laws of logarithms}}}

If $x$ and $y$ are positive numbers and $r$ is a rational number, then

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

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

\item[3.]  $\ln (x^{r})=r\,\ln \,x$
\end{enumerate}

\subsubsection{Exponential Function Natural Base\label%
{exponential function natural base}}

$e$ is the number such that $\ln \,e=1$.

\subsubsection{\QTR{BOXHEAD}{Properties of the Exponential Function\label%
{properties of the exponential function}}}

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 
\[
\lim\limits_{x\rightarrow -\infty }\,\,e^{x}=0\qquad \quad
\lim\limits_{x\rightarrow \infty }\,\,e^{x}=\infty 
\]
so the $x$-axis is a horizontal asymptote of $f(x)=e^{x}$.

\subsubsection{\QTR{BOXHEAD}{Laws of Exponents Base }$\QTR{BOXHEAD}{e}$ 
\QTR{BOXHEAD}{\label{laws of exponents base e}}}

If $x$ and $y$ are real numbers and $r$ is rational, then

\begin{enumerate}
\item[1.]  $e^{x+y}=e^{x}e^{y}$

\item[2.]  $e^{x-y}=\dfrac{e^{x}}{e^{y}}$

\item[3.]  $(e^{x})^{r}=e^{rx}$
\end{enumerate}

\subsubsection{\QTR{BOXHEAD}{Laws of Exponents, General\label%
{laws of exponents, general}}}

If $x$ and $y$ are real numbers and $a$, $b>0$, then

\begin{enumerate}
\item[1.]  $a^{x+y}=a^{x}a^{y}$

\item[2.]  $a^{x-y}=a^{x}/a^{y}$

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

\item[4.]  $(ab)^{x}=a^{x}b^{x}$
\end{enumerate}

\subsubsection{Exponential Limit\label{exponential limit}}

$\lim\limits_{x\rightarrow 0}\,\,(1+x)^{1/x}=e$

\subsection{Related Scientific Notebook Topics}

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

{\small \hyperref{Laws of exponents, computing the laws}{}{}{%
../../Help/DM3-5.tex#Exponential function}}

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

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

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