1. Slide 4.4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rules of Logarithms Learn the rules of logarithms. Learn to change the base of a logarithm. Learn to apply logarithms in growth and decay. Learn to apply logarithms in carbon dating. SECTION 4.4 1 2 3 4

3. Slide 4.4- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RULES OF LOGARITHMS Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. The logarithm of the product of two (or more) numbers is the sum of the logarithms of the numbers. 1. Product Rule

4. Slide 4.4- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RULES OF LOGARITHMS Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers. 2. Quotient Rule

5. Slide 4.4- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RULES OF LOGARITHMS Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. The logarithm of a number to the power r is r times the logarithm of the number. 3. Power Rule

6. Slide 4.4- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Rules of Logarithms to Evaluate Expressions Given that log 5 z = 3 and log 5 y = 2, evaluate each expression. Solution

7. Slide 4.4- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Rules of Logarithms to Evaluate Expressions Solution continued

8. Slide 4.4- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Rules of Logarithms to Evaluate Expressions Solution continued

9. Slide 4.4- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Writing Expressions In Expanded Form Write each expression in expanded form. Solution

10. Slide 4.4- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Writing Expressions In Expanded Form Solution continued

11. Slide 4.4- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Writing Expressions in Condensed Form Write each expression in condensed form.

12. Slide 4.4- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Writing Expressions in Condensed Form Solution

13. Slide 4.4- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Writing Expressions in Condensed Form Solution

14. Slide 4.4- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Writing Expressions in Condensed Form Solution

15. Slide 4.4- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CHANGE-OF-BASE FORMULA Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then log b x can be converted to a different base as follows:

16. Slide 4.4- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Using a Change of Base to Compute Logarithms Compute log 5 13 by changing to (a) common logarithms and (b) natural logarithms. Solution

17. Slide 4.4- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley LOGARITHMS WITH EXPONENTIAL BASE Here’s an example where k = –1.

18. Slide 4.4- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Evaluating an Expression Find the value of each expression without using a calculator. Solution

19. Slide 4.4- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Evaluating an Expression Solution continued

20. Slide 4.4- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Matching Data to an Exponential Curve Find the exponential function of the form f (x) = ae kx that passes through the points (0, 2) and (3, 8). Solution Substitute (0, 2) into f (x) = ae bx. So a = 2 and f (x) = 2e kx. Now substitute (3, 8) in to the equation.

21. Slide 4.4- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Matching Data to an Exponential Curve Solution continued Now solve for b. Thusis the desired function.

22. Slide 4.4- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MODEL FOR EXPONENTIAL GROWTH OR DECAY A(t) = amount at time t A 0 = A(0), the initial amount k = relative rate of growth (k > 0) or decay (k < 0) t = time

23. Slide 4.4- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Calculating Carbon Emissions in the United States and China In 1995, the United States emitted about 1400 million tons of carbon into the atmosphere (about one-fourth of worldwide emissions). In the same year, China emitted about 850 million tons. Suppose the annual rate of growth of the carbon emissions in the United States and China are 1.5% and 4.5%, respectively. After how many years will China be emitting more carbon into Earth’s atmosphere than the United States will?

24. Slide 4.4- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Calculating Carbon Emissions in the United States and China Solution Let t = 0 correspond to 1995, then Find t so that That is, solve for t:

25. Slide 4.4- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Calculating Carbon Emissions in the United States and China Solution continued So, in less than 17 years from 1995 (around 2012), under the present assumptions, China will be emitting more carbon into the Earth’s atmosphere than the U.S. will.

26. Slide 4.4- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Carbon Dating A human bone in the Gobi desert is found to contain 30% of the carbon-14 that was originally present. (There are several methods available to determine how much carbon-14 the artifact originally contained.) How long ago did the person die? Solution The half-life of carbon-14 is approximately 5700 years and that means

27. Slide 4.4- 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Carbon Dating Solution continued Substitute this value for k.

28. Slide 4.4- 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Carbon Dating Solution continued Since the bone contains 30% of the original carbon-14, we have, The person died about 9900 years ago.

29. Slide 4.4- 29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 King Tut’s Treasure In 1960, a group of specialists from the British Museum in London investigated whether a piece of art containing organic material found in Tutankhamun’s tomb had been made during his reign or (as some historians claimed) whether it belonged to an earlier period. We know that King Tut died in 1346 B.C. and ruled Egypt for 10 years. What percent of the amount of carbon- 14 originally contained in the object should be present in 1960 if the object was made during Tutankhamun’s reign?

30. Slide 4.4- 30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 King Tut’s Treasure Solution The half-life of carbon-14 is approximately 5700 years and that means Solving for k yields k = –0.0001216. Substituting this value into the equation yields,

31. Slide 4.4- 31 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 King Tut’s Treasure Solution continued The time t that elapsed between King Tut’s death and 1960 is t = 1960 + 1346 = 3306. The percent x 1 of the original amount of carbon- 14 remaining after 3306 years is Let x represent the percent of the original amount of carbon-14 in the object that remains after t yrs.

32. Slide 4.4- 32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 King Tut’s Treasure Solution continued King Tut ruled Egypt for 10 years, the time t 1 that elapsed from the beginning of his reign to 1960 is t 1 = 3306 + 10 = 3316. The percent x 2 of the original amount of carbon- 14 remaining after 3316 years is Thus, if the piece of art was made during King Tut’s reign, the amount of carbon-14 remaining in 1960 should be between 66.816% and 66.897%.