Assignment 61

 "The most distinct and beautiful statements of any truth must take at last the mathematical form." -- (Henry David Thoreau, 1817-1862) 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. Previous Tidbits have referenced the digits 1,2,3,4,5,6,7,8,9. Here are just brief statements about some other numbers. 10: The number 10 = 1+2+3+4. Since the ancient Greeks saw "four-ness" in many things, they considered 10 the ideal number. 17: Not notably important in Christian tradition, but quite important in the ancient Near East. The Muslim alchemist Jabir ibn Hayyan saw the entire material world based on 17, since he claimed the series of numbers 1,3,5,8 (which sum to 17) formed the foundation of all other numbers 20: This number formed the basis for counting in some ancient cultures. After all, if we count our fingers and toes, we get 20. The Maya used base 20. 33: A number of completion and perfection. In Christian tradition, Jesus lived on earth for 33 years, and David ruled for 33 years. 40: Of the double digit numbers, 40 is one of the most fascinating. It appears quite frequently in history. Lent lasts for 40 days, the Biblical flood lasted 40 days, the 40 large stone pillars in Stonehenge are arranged in a sacred circle with a diameter of 40 steps, the children of Israel wandered in the desert for 40 years, etc. What did Herkimer learn when he tried to light a fire in his kayak? Answer: You can't have your kayak and heat it too. Herky's friends: JACK O'LANTERN...this guy worked in a pumpkin patch. AL E. BYE... he had an excuse for everything. ASSIGNMENT #61 Reading: Section 8.5, pages 493-495 Written: Pages 496-497/15,17,19,35-55(odds),61,69.

 Mathematical word analysis:RANDOM: From the French root randir, which means "to gallop." Perhaps the best explanation is the thought that when a horse is in full gallop the rider has little on no control.
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.".

Ln(x) is simply Logbx. Ln simply means that the base is e. If you use base e, you are working with natural logarithms. If you use base 10, you are working with common logarithms.

Your calculator has only two log functions. One with base 10 (LOG) and one with base e (LN). However, you can find logarithms for any legal base by using what I like to call the Crazy Base Theorem, a description offered by Allan Gunther, former great math teacher at Cate. (Most books call this theorem the Change-of-Base Theorem.). This very useful theorem says

logbA = logcA/logcb

Suppose, for example, you want log5498. Our calculator doesn't have base 5, but the Crazy Base Theorem lets us calculate this by using base 10 or base e. The value is log(498)/log(5) = 3.858862792. If you raise 5 to the power 3.858862792, you will get 498. We could have used base e and and calculated ln(498)/ln(5). Try it!

Let's prove the Crazy Base Theorem. (It certainly doesn't represent an obvious statement.)

If logcA = x and logcb = y, then cx = A and cy = b.

It follow that c = A1/x and c = b1/y, which implies that A1/x = b1/y ==> A = bx/y ==> x/y = logbA.

Here's an example that makes use of the Crazy Base Theorem. If I want to know how long it will take money to double at a true rate of 6% a year, I must solve (1.06)x = 2. OK, we have x = log1.062 = log(2)/log(1.06) = 11.89, or about 12 years.

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Properties of logs (These can be summarized in four statements):

log(AB) = logA + logB (When you multiply numbers with like bases, you add exponents: 109x106 = 1015)

log(A/B) = logA - logB (When you divide numbers with like bases, you subtract exponents: 109/106 = 103)

logAN = NlogA (You multiply exponents in a situation like (109)6 = 1054)

logbA = logcA/logcb (Crazy Base Theorem)

Problem example: Express log(AB2/C3) as a sum and/or difference of logs.

Solution: logA + logB2 - logC3, which could be written logA + 2logB - 3logC.

Problem example: Write 4logX + 6logY as a single logarithm.

Solution: logX4 + logY6 = log(X4Y6).

 Problem: Evaluate log3275. Solution: (with communication): log3275 = 5(log327) = 5(3) = 15. Problem: Evaluate log13987. Solution (with communication): Using the Crazy Base Theorem, we have log13987 = log(987)/log(13) = 2.688. Problem: Write log x7 - log y2 + log w1/2 as a single logarithm. Solution (with communication): log x7 - log y2 + log w1/2 = log(x7/y2) + log÷(w) = log[x7÷(w)/y2]. Problem: Solve for x: ln 15 + 2ln x = ln 135. Solution (with communication): ln 15 + 2ln x = ln 135 ==> ln 15 + ln x2 = ln 135 ==> ln x2 = ln 135 - ln 15 = ln(135/15) = ln 9 ==> x2 = 9 ==> x = 3 or x = -3. Since ln x is defined only for x > 0, the only solution to the original equation is x = 3.