  Assignment 60

 "Statistics is the logic of measurement, and all sciences require measurement." -- (S. M. Stigler, 1941 - ) Numbers have interesting historical backgrounds, and there are many interesting stories behind just about any counting number. Here are just a few very brief tidbits relating to numbers. (Some research will yield many interesting stories.) 8. Like most other single digit numbers, 8 has religious and mystical associations, but the ancients considered it interesting for purely mathematical reasons. It is closely associated with music, with 8 notes in an octave. The Greeks discovered that every odd number greater than 1, when squared, results in a multiple of 8, plus 1. They also discovered that squares of odd numbers greater than 1 differed from each other by a multiple of 8. 9. The number 9 has been associated with both negative and positive things. The Chinese, Mongols, and Turkic peoples have been very fond of 9. On the other hand, some Christian writings associate it with pain and sadness; for example, the Ninth Psalm contains a prediction of the Antichrist. From a purely mathematical standpoint, it is the only square number that is the sum of two consecutive cubes. (13 + 23 = 9), and a number can be evenly divided by 9 only if its digits add up to a multiple of 9. Why does Herkimer think that lions are very religious? Answer: Because they prey a lot. Herky's friends: LAUREL N. HARDEE..she was a great fan of a comedy team from years ago. PHIL TRATION...he sold water purifiers. ASSIGNMENT #60 Reading: Section 8.4, pages 486-489. READ CAREFULLY. Written: Page 490/3-14. Don't be mechanical. THINK! Mathematical word analysis:LOGARITHM: This word comes from a combination of the Greek roots logos (ratio) and artihmus (number). John Napier used the ratio concept when he "invented" logarithms. (See more on Napier below.)
Characterization: A log is an exponent. If you keep this in mind, working with logarithms becomes sensible and very useful. The statement y = logbx is read "y is the power to which b must be raised to get x." For instance, log232 is the power to which 2 must be raised in order to get 32. Since 25 = 32, it follows that log232 = 5.

y = logbx <==> by =x

Your calculator has a LOG key. This function uses the base 10. For instance, if you use the calculator to compute log(100), it will produce 2, since 2 is the power to which 10 must be raised in order to get 100.

If you calculate log(8425), you will get 3.92556991. If you now calculate 10^(3.92556691), you will get 8425. Note that the 3.92556991 is the power to which you must raise 10 in order to get 8,425.

The other log function on your calculator is LN. This function uses the base e. If you calculate ln(8425), you will get 9.038958755. If you now calculate e^(9.038958755), you will get 8,425.

Here is the MATH HISTORY TIDBIT from Assignment #50. It relates to logarithms.

John Napier (1550-1617): A native of Scotland, Napier invented logarithms as a laborsaving device to speed up numerical computation processes. His invention was enthusiastically adopted throughout Europe. Today we consider a logarithm as an exponent, but Napier never thought of his discovery in this way. The modern use of logarithms in high-powered mathematics course and in computer technology would probably boggle Napier's mind. In 1614 Napier published a book titled A Description of the Wonderful Law of Logarithms. It was enthusiastically accepted throughout Europe as the use of the tables it contained enabled tedious multiplication and division operations to be done as easily as addition and subtraction. Interestingly, Napier, like Michael Stifel (1486-1567) before him, was a number mystic, and spent time attempting to determine a date upon which the world would end.

 Problem: Solve for x: log381 = x. Solution (with communication): log381 = x ==> 3x = 81 = 34 ==> x = 4. Problem: Solve for w: logw16 = 2. Solution (with communication): logw16 = 2 ==> w2 = 16 ==> w = 4. Note carefully: If you solve w2 = 16, then w = -4 is also a solution. However, the base of a logarithm cannot be negative. Problem: Solve for x: ln(x) = 7. Solution (with communication): ln(x) = 7 ==> x = e7 = 1096.93. (Answer expressed to 2 decimal places.) Problem: (a) Solve for x: log(x-12) = -3. (b) Find the domain and range of the function y = log(x - 12). Solution (with communication): (a) log(x-12) = -3 ==> x-12 = 10-3 ==> x = 12 + 10-3 = 12.001. (b) We must have x - 12 > 0. Hence, the domain of the function is all real numbers greater than 12. The variable y can assume any real number value, so the range of the function is all real numbers. NOTE: Don't get confused here. You can't take the log of a negative number, but a log can be negative. For instance, log(-2) is not defined since -2 is not in the domain of y =log x. However, the equation log x = -2 does have a solution, which is x = 10-2 = 1/100.