\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) Enter the number 5 in a
calculator showing 10 decimal digits after the decimal point. Press
the square root button 20 times. The result will be {\bf
1.00000\thinspace 15348}\thinspace. Subtract 1 and multiply by
1,048,576 to get {\bf 1.60943\thinspace 91475} {\it but}\/ the same
calculator will declare that $\ln 5$ is {\bf 1.60943\thinspace
79124}. Since 1,048,576 is $2^{20}$, this is {\it not}\/ a
concidence. Explain.

\medskip

\noindent b) Given a positive number, $x$, outline a strategy for
computing $\ln x$ only with the arithmetic operations ($+$, $\times$,
$-$, $/$) and square root (\raise
.2em\hbox{$\sqrt{\hphantom{zi}}$}). Your strategy should involve
asserting (and verifying) that a certain sequence which can be easily
computed with the listed operations always converges to $\ln x$.

\vfil\eject\end