Mathematics 131 – Calculus for Physical and Life Sciences 1 Discussion 1 – Irrational, Impossible Relations Friday, September 14, 2007

Goals

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.

Discussion Questions

Some “warmups” first:

A) If log_{2}(x) = s, then what is log_{1/2}(x)? B) If log_{b}(x) = s, then what is log_{1/b}(x)? (Assume b > 0 of course!) C) If log_{b}(x) = s, then what is log_{b^{2}}(x)?

Now to some more interesting questions.

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.

1) Say log_{2}(3) = r. Then by the definition, 3 = 2^{r}. Similarly, what number x satisfies 2 = e^{x}?

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^{something} = 3.

3) Now solve the equation from part 2 for the “something” and use that to determine a formula for r = log_{2}(3) 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. (Answer: Approximately, log_{2}(3) = 1.584962501 · · ·.)

4) Similarly, use the TI-30 calculator provided to compute log_{5}(27).

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.

F) Let’s try some mathematics to see if we can find the answer.

1) Assume that log_{2}(3) = a/b for some integers a, b. Show that a, b must satisfy 2^{a} = 3^{b}.

2) Notice that a = b = 0 makes this equation true. Does that help us, though?

3) Find a ≠ 0 and b ≠ 0 such that 2^{a} = 3^{b} or show that no such a, b exist. Is log_{2}(3) rational or irrational?

4) (Extra Credit) Assume that p, q are any two integers with no common integer factors. Is log_{p}(q) rational or irrational?