Wednesday, November 28, 2018

Logarithms: How Math Helps Science

I'm teaching my algebra students about logarithms today. It is likely the hardest algebra topic there is. When I started at Contra Costa College, after having taught full-time at Muskegon Community College for 6 years, I think I still was not 100% solid on logs. (I could do the problems, but proving those properties was a tangled mess.)

I had developed my murder mystery project because my officemate at MCC was a chemistry teacher, and she had complained that teachers weren't teaching logs well enough in Intermediate Algebra.I hoped that project would help. I think it does, but I don't have anything solid to base that on.

For many years now, I've told students a bit of history in my introduction to logarithms. John Napier invented them in 1614 as a way of making hard calculations easier. (Can you imagine finding the square root of 192.7 without a calculator? You'd have to guess, check, and modify repeatedly...)

Around that time Kepler came up with his three laws of planetary motion. Over the years, I have repeatedly wondered out loud whether Kepler was using logarithms while he figured these things out.
Today when this came up, I was able to turn to our tutor and ask him to look it up. He found a fascinating article on it.

Turns out Kepler came up with the first two "laws" before playing with Napier's new idea. But his third came after.

I. Planets move in ellipses with the Sun at one focus.
II. The radius vector (line from sun to planet) describes equal areas in equal times.
III. "The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances."

This is an improper use of the word proportion. He was making a proportion from the logs of the ratios, and saw that log(T1/T2) = 1.5*log(r1/r2). In modern terms, we say something like: The square of the ratio of the times (that it takes two planets to go around the sun) equals the cube of the ratio of their mean distances from the sun. The fact that he was thinking in terms of logs shows how helpful the new (in 1614) concept was for helping scientists see patterns. (Read that article for more on Kepler and Napier.)

I love that math can help us see new things.

1 comment:

1. Newton-Raphson (which was known long before Newton or Raphson) makes figuring square-roots by hand not too bad. 192.7 is close to 14^2 (196), so the 2nd estimate would be 14 - (14^2-192.7)/28 = 14-33/280 = 3887/280. This turns out to be about 0.0005 off of the actual square root of 192.7. It converges quadratically, so the number of accurate digits roughly doubles with each iteration.

I recently came across a problem that was of the form (1/log_a x) + (1/log_{a^2} x) + ... + (1/log_{a^k} x) = C, with the challenge to solve for x. That becomes much easier if you know the identity log_x y = 1/log_y x. The LHS becomes log_x a + log_x a^2 + ... + log_x a^k = log_x a^{1+2+...+k} = (k(k+1)/2) log_x a, which means that log_a x = k(k+1)/(2C). With proper choice of a, k and C this problem can look hard, but work out really easy.