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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Exponential and Logarithmic functions}


\subsubsection{Basic definitions }

Exponential functions are functions of the sort \( f(x)=a^{x} \). \( a \)
is called the \emph{base} of the exponential., and has to be positive. They
are distinctly different from power functions, although students often confuse
them. For example, if \( a=2 \), the function \( f(x) \) satisfies \( f(0)=1 \),
\( f(1)=2 \), \( f(2)=2 \), \( f(3)=8 \), \( f(-2)=1/4 \). Here is a great
plot of various exponential functions (various bases, that is). 

\vspace{0.3cm}
{\par\centering \includegraphics{exponentials.ps} \par}
\vspace{0.3cm}

The various color graphs correspond to various bases. The green graph is the
plot of \( (1/2)^{x} \), the red one (kind of silly) is the plot of \( 1^{x} \),
the yellow one, \( 2^{x} \), the light blue one, \( e^{x} \), the dark blue
one, \( 3^{x} \), and, finally, the violet one, \( 10^{x} \). 


\subsubsection*{Rules}

There are only two basic algebraic rules for exponentiation. They are, when
you think about them in terms of integer powers, obvious.

\begin{prop}
For all numbers \( a,\: b,\: \textrm{and}\: c \), \begin{eqnarray*}
a^{b}a^{c} & = & a^{b+c},\: \textrm{and}\\
\left( a^{b}\right) ^{c} & = & a^{bc}.
\end{eqnarray*}

\end{prop}
The primary reason to study exponential functions is that they model many natural
phenomena, that is, the behavior of many systems are expressed in terms of exponential
functions. Such things as compound interest, radioactive decay, and population
growth are governed by exponential functions.

One curious number always mentioned in this context is \( e \). \( e \) is
just a number, like \( \pi  \), which has a specific value. That letter, \( e \),
is almost-always reserved for that particular number (again, like \( \pi \approx 3.14159 \).
\( e \) is approximately \( 2.71828 \). But \( e, \) like \( \pi  \), comes
from a specific relationship. If you look at the graphs of the various exponential
functions \( y=a^{x} \) for a number of choices of \( a \), they all pass
through \( (1,0) \), since \( a^{0}=1 \) for any positive number \( a \).
\( e \) is that choice of base so that the slope of the curve as it passes
through \( (1,0) \) is 1. 

By the way, just like \( \pi  \), \( e \) is irrational.


\subsection*{Logarithms}

Be sure to read throught the general material about inverse functions.

Logarithms are the inverse of exponentiation, that is, \( \log _{a}x \) is
the inverse of \( f(x)=a^{x} \). Said another way, logarithms are just exponents
in reverse: \[
y=\log _{b}x\, \Leftrightarrow \, x=b^{y}.\]
So, \( \log _{4}2=1/2 \), \( \log _{3}(1/3)=-1 \), and so on. There is a special
case, when the base is \( e \). Then, instead of writing \( \log _{e}x \),
we write \( \ln x \), the \textit{natural} logarithm of \( x \). There is
also another notation for \( \log _{10}x \). It is sometimes called \( \log x \),
the \textit{common} logarithm of \( x \). Unfortunately, most post-calculus
math texts now confuse the notational issue even more, by writing the \textit{natural}
logarithm of \( x \) as \( \log x \). Hopefully, it will be clear from the
context which is meant. To a mathematician, there is no special significance
of the number 10, except for the number of fingers we have. 


\paragraph*{Rules for Logarithms:}

These are just rules for exponents, \textit{backwards}. In fact, the logarithm
functions are precisely the inverse of exponentials: \[
\log _{b}(b^{x})=x\, \mathrm{and}\, b^{\log _{b}(x)}=x,\]
for all bases \( b \) and all values of \( x \) for which the expressions
make sense. In particular, \( \ln (e^{x})=x \), and \( e^{\ln x}=x \).

\begin{prop}
For all \( b \), \( c \), and \( d \), \( c \) and \( d>0 \),   
\begin{enumerate}
\item \( \log _{b}(cd)=\log _{b}(c)+\log _{b}(d) \), 
\item \( \log _{b}(c^{d})=d\log _{b}(c) \), Here \( d \) can be negative. 
\item \[
\frac{\log _{b}(c)}{\log _{b}(d)}=\log _{d}(c).\]
 This last fact means that you can find the logarithm of \( c \) to any base,
if you know all logarithms of one base.
\end{enumerate}
\end{prop}
\textbf{\emph{Proof:}}

Here are some equations involving logarithms:

\begin{example}
\( \log _{2}80-\log _{2}5=\log _{2}(80/5)=\log _{2}16=4 \).
\end{example}
%

\begin{example}
\( 2^{x}=e^{x\, \ln 2} \).
\end{example}

\subsubsection*{Solution}

\begin{xca}
Solve for \( x \): \( \ln x+\ln (x-1)=0 \).
\end{xca}
\[
x=\]


The graphs of the various logarithm functions, for different bases \( b \),
are  shown below. The bases used are \( b=2 \), \( e \), \( 3 \), and 10.

\vspace{0.3cm}
{\par\centering \includegraphics{logarithms.ps} \par}
\vspace{0.3cm}

The thing I want you to see with these plots is the fact that they all pass
through the point \( (1,0) \), but with different slopes. The steepest is that
of \( \log _{2}(x) \), and the shallowest (at that point) is \( \log _{10}(x) \).
\end{document}
