  Assignment 62

 "Courage lies in the wanting to know, which solves nothing, but which has within itself all solutions." -- (Francoise Mallet-Joris) 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, and a few other numbers. Here are brief statements about some other numbers. 108: Tibetan sacred scriptures, the Tanjur and the Kanjur, contain 108 parts. 120: This number is connected with lifespan in Old Testament (Genesis 6:3) 153: The disciples of Christ catch exactly 153 fishes (John 21:11) 248: This number has a prominent place in Judaism as it represents the numerical value of the first two words of the fundamental religious expression of monotheism, "Hear, O Israel." 666: The number of the Beast (Revelation 13:8) in the Christian Bible. What did Herkimer say to the noisy people who lived in the apartment below his? Answer: Do under others as you would have them do under you. Herky's words of wisdom: Read contracts carefully. The fine print may be a clause for suspicion. Make sure your dentist is honest. When he does an extraction, you should make sure you get the tooth, the whole tooth, and nothing but the tooth. ASSIGNMENT #62 Reading: Section 8.6, pages 501-504. Writing: Pages 505-506/25,31,34,39,50,52, 63,64,65 Mathematical word analysis:INCH: From the Latin unica ("twelfth part"). An inch is one of twelve parts that make up the unit measure we call 1 foot.
Here are two examples of solving logarithmic equations.

Example 1:

log3(x-1)2 = 2 ==> (x-1)2 = 9 ==> x-1 = 3 or x-1 = -3 ==> x = 4 or x = -2. Checking answers, both work.

Example 2:

log2(x+1) - log2(x-1) = 3 ==> log2[(x+1)/(x-1)] = 3 ==> (x+1)/(x-1) = 23 = 8 ==> x+1 = 8x-8 ==> 7x = 9 ==>x = 9/7. Checking this answer, it works.

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 in Cate terminology. It is commonly called the Change-of-Base Theorem.)

 Problem: Solve for x: 84x = 64x-9. Solution (with communication): 84x = 64x+9 ==> 84x = (82)x+9 ==> 84x = 82x+18 ==> 4x = 2x+18 ==> 2x=18 ==>x = 9. Problem: Solve for z: 72z - 34 = 693. Solution (with communication): 72z - 34 = 693 ==> 72z = 693 + 34 = 727 ==> 2z = log7727 = log(727)/log(7) ==> z = (1/2)log(727)/log(7) = 1.693. Problem: Solve for w: ln w - ln 12 = 4. Solution (with communication): ln w - ln 12 = 4 ==> ln(w/12) = 4 ==> w/12 = e4 ==>w = 12e4 = 655.178. Problem: A substance decays at a rate of 2.3% a year. How many years will pass before only 25% of the original amount remains? Solution (with communication): If x is the requested number of years, then (1-0.023)x = 0.25 ==>(0.977)x = 0.25 ==> x = log(0.25)/log(0.977) = 59.578. It will take approximately 60 years to lose 75% of the original amount.