Assignment 65

"Fate laughs at probabilities." - (Lytton Bulwer)

Math History Tidbit:


Abacus: Dating back perhaps 5,000 years, this device involving rods and beads was used by the Chinese, Greeks, and Romans. It is still used in some parts of the world today.

Napier's bones: Invented by John Napier (Scotland, 1550-1617), this simple device consisted of a series of rods containing the digits 1 to 9. Rotating the rods facilitated multiplication.

Slide rule: Napier's discovery of logarithms led to the invention of this device by William Oughtred (1574-1660). It is really just a compact set of logarithm tables, and it was widely used well into the twentieth century.

Herkimer's Corner

Where did Herkimer leave his pet pigs while he took a stroll in a big city?

Answer: In a porking lot.

Herky wants to know:

Why should there be chicken in chicken soup when there is no horse in horseradish?

If you believe in the motto "the customer is always right," what do you do when a customer insists he was wrong?


Reading: Review Chapter 8, as necessary.

Writing: The review problems below (in Items for reflection.)


Items for reflection:

Mathematical word analysis:
PRISM: From the French priein ("to saw"). A prism is a solid whose bases are similar, and parallel polygons. It is a form that could be created (sawed) by a craftsman.


1. Using the laws of logarithms, write each expression as a single logarithm:

(A) logA + logB -logC
(B) log5x + log5y6
(C) (1/2)logx + (1/4)logy

2. Evaluate each expression. You should be able to do each one without a calculator.

(A) ln e7

(B) log232

(C) 8log83

(D) log1099

(E) log1/327

(F) log4(0.25)

(G) log 10003

(H) log(1/1000)

3. State the domain and range for the function y = log7(x-12)?

4. Solve each equation. (Remember, you should check for extraneous solutions since log is a restricted function.)

(A) log3(4x) = log3(x+15)

(B) -182 + 52x-7 = 46

(C) 12+ e7x = 5

(D) 85/(2 + 6e-3x) = 5

5. Growth and Decay problems:

(A) The value of a new car is $23,200. If this particular car model loses 11% of its value each year, when will it be worth half of its purchase price?

(B) How many years will it take $1,000 to grow to $6,000 is the interest rate is

(i) a true rate of 7% ?
(ii) 7% compounded quarterly?
(iii) 7% compounded continuously?

(C) A series EE U.S. Savings Bond can be purchased for $18.75, and redeemed at a later date for $25. What is the annual yield if the bond can be redeemed after 5.5 years?

(D) A series EE U.S. Savings bond can be purchased for $18.75 and redeemed at a later date for $25. If the government wants the annual yield to be 5.8%, what should be the time limit between purchase and redemption?

6. If f(x) = y = 5 + log12(x+5), then f-1(x) =

7. The table gives the mean weight w (in kilograms) and age x (in years) of Atlantic cod from the Gulf of Maine. Check to see if an exponential model is a good fit for the data.



















Problem: Solve for x: log7(x+8) = log7(x) + 2.

Solution (with communication):

log7(x+8) = log7(x) + 2

==> log7(x+8) - log7(x) = 2

==> log7[(x+8)/x] = 2

==> (x+8)/x = 72 = 49

==> x + 8 = 49x

==> 48x = 8

==> x = 1/6.

Problem: How long will it take a single deposit to increase 20-fold if the annual interest rate is 18%?

Solution (with communication):

If T is the requested number of years, then

(1.18)T = 20

==> T = log1.1820 = log(20)/log(1.18)

==> T = 18.0995.

Rounding up, it would take 19 years for $1 to grow to $20 if the annual interest rate is 18%.