**Kevin, age 9.7, finds identities for logarithms and graphs
y = log _{10}x**

Don wrote the first column of logs below, then asked Kevin to guess and write down what the log of 1 is (of course he had no idea what a log is!!). Then he was to find the log of 1 on a calculator (0). Then guess and write down the log of 2, then write down what the calculator gives for the log of 2 (0.301) and do the same for the rest of these. They did each to 3 decimal places:

**Then
Don asked Kevin to look for a pattern in the numbers**.
Kevin saw that 2 _{*} log 2 = log 4 and 3 _{*} log
2 = log 8 and Don showed Kevin he could write 8 as 2^{3
},
so
he wrote 3 _{*} log 2 = log 2^{3 }and Don asked him to do
more like these and generalize. See his first generalization on the graph. With
Don's egging him on, he found log 2 + log 3 = log 6 = log (2x3) and he did more
like this and generalized it on the graph.

Kevin took about 2 -45 minute sessions to do this. He did a fine job! Don asked if he would look for more patterns.

Don and Kevin talked about writing the log equation log_{10}100
= 2
(read as the log of 100 base 10 equals 2), as the
exponential equation 10^{2} =
100. Then Don asked kevin what the log_{10}200
= ? He wasn't sure, but Don said he could use his identity above log_{10}200
= log_{10}(2x100) = log_{10}2
+ log_{10}100.
He knew log_{10}2 =
0.301 and log_{10}100
= 2,
so log_{10}2
+ log_{10}100 =
0.301 + 2 = 2.301 !! **EASY. **Kevin then found
log_{10}300, log_{10}400,
up to log_{10}1000. Then
after a little discussion about the scales, he graphed y= log_{10}x
from 0 to 1000, by 100's for x. Kevin said he would mail this graph to Don when
he finished it. Along the way, Don told Kevin about how the logarithm was
invented by Napier (along with some others), and the result was to vastly
increase the computational powers of astronomers like Tycho Brahe and Kepler.
That was because products are changed into sums using Kevin's
identity log_{10}A
+ log_{10 }B = log_{10}(AxB)
and when one is dealing with large numbers, as with distances to the moon and
sun, it it much simpler to add the logs, than multiply the numbers.

To other discoveries

To download Don's materials

Mathman home