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\centerline{Mathematics 131 -- Calculus for Physical and Life Sciences 1}
\centerline{Discussion 1 -- Irrational, Impossible Relations}
\centerline{Friday, September 14, 2007}
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{\it Goals} 
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The following questions are intended to solidify your understanding
of logarithms, get you familiar with the basic scientific calculators
that you will use on exams in this course,  and lead you eventually 
into some interesting mathematics
about which numbers are irrational.  You will be working on 
these questions in a group with three other students since
``the more heads, the better'' here.  
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{\it Discussion Questions}
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Some ``warmups'' first:
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\item{A)}  If $\log_2(x) = s$, then what is $\log_{1/2}(x)$?
\item{B)}  If $\log_b(x) = s$, then what is $\log_{1/b}(x)$?  (Assume $b > 0$ of course!)
\item{C)}  If $\log_b(x) = s$, then what is $\log_{b^2}(x)$?
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Now to some more interesting questions.  
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\item{D)} The TI-30 calculators that 
we use on exams have only $\log$ (log base 10) and $\ln$ keys.
Suppose we need to compute a value like $\log_2(3)$.  How can we
do this?  This question will lead you to a method.
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\itemitem{1)}  Say $\log_2(3) = r$.  Then by the definition,
$3 = 2^r$.  Similarly, what number $x$ satisfies $2 = e^x$?
\itemitem{2)}  Taking your $x$ from part 1, substitute $e^x$ for the $2$ in the
equation $2^r = 3$ to get a new equation of the form $e^{\rm something} = 3$.
\itemitem{3)}  Now solve the equation from part 2 for the ``something'' 
and use that to determine a formula for $r = \log_2(3)$ {\it using only
values of the natural log function}.  This means that you can compute
a value for $\log_2(3)$ using our calculators now!  Do this.
({\it Answer:}  Approximately, $\log_2(3) = 1.584962501\cdots $.)
\itemitem{4)}  Similarly, use the TI-30 calculator provided
to compute $\log_5(27)$.  
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\item{E)}  Is it possible to tell whether these numbers are rational or 
irrational from the information you get from a calculator?  Explain,
recalling that the decimal expansion of a rational number is always
eventually repeating or terminating, but the repeating pattern can be 
any pattern of digits.
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\item{F)}  Let's try some mathematics to see if we can find the answer.
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\itemitem{1)}  Assume that $\log_2(3) = a/b$ for some integers $a,b$.
Show that $a,b$ must satisfy $2^a = 3^b$.
\itemitem{2)}  Notice that $a = b = 0$ makes this equation true.
Does that help us, though?
\itemitem{3)}  Find $a\ne 0$ and $b \ne 0$ such that $2^a = 3^b$ 
or show that no such $a,b$ exist.   Is $\log_2(3)$ rational or irrational?
\itemitem{4)}  (Extra Credit) Assume that $p,q$ are any two integers with no common integer
factors.  Is $\log_p(q)$ rational or irrational? 
\bye
