1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 4 Exponential and Logarithmic Functions

2. OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Rules of Logarithms Learn the rules of logarithms. Change the base of a logarithm. Apply logarithms in growth and decay. SECTION 4.4 1 2 3

3. 3 © 2010 Pearson Education, Inc. All rights reserved 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. 4 © 2010 Pearson Education, Inc. All rights reserved 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. 5 © 2010 Pearson Education, Inc. All rights reserved 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. 6 © 2010 Pearson Education, Inc. All rights reserved 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. 7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using Rules of Logarithms to Evaluate Expressions Solution continued

8. 8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using Rules of Logarithms to Evaluate Expressions Solution continued

9. 9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Expressions In Expanded Form Write each expression in expanded form. Solution

10. 10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Expressions In Expanded Form Solution continued

11. 11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Writing Expressions in Condensed Form Write each expression in condensed form.

12. 12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Writing Expressions in Condensed Form Solution

13. 13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Writing Expressions in Condensed Form Solution continued

14. 14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Writing Expressions in Condensed Form Solution continued

15. 15 © 2010 Pearson Education, Inc. All rights reserved CHANGE-OF-BASE FORMULA Let a, b, and x be positive real numbers with a ≠ 1 and b ≠ 1. Then log b x can be converted to a different base as follows:

16. 16 © 2010 Pearson Education, Inc. All rights reserved 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. 17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Matching Data to an Exponential Curve Find the exponential function of the form f (x) = ae bx that passes through the points (0, 2) and (3, 8). Solution Substitute (0, 2) into f (x) = ae bx. So a = 2. Now substitute (3, 8) into the equation.

18. 18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Matching Data to an Exponential Curve Solution continued Now solve for b. Thusis the desired function.

19. 19 © 2010 Pearson Education, Inc. All rights reserved STANDARD GROWTH FORMULA 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 Exponential growth (or decay) occurs when a quantity grows (or decreases) at a rate proportional to its size. The standard growth formula is

20. 20 © 2010 Pearson Education, Inc. All rights reserved HALF-LIFE FORMULA The half-life of any quantity whose value decreases with time is the time required for the quantity to decay to half its initial value. The half-life of a substance undergoing exponential decay at a rate k (k < 0) is given by the formula

21. 21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Finding the Half-Life of a Substance In an experiment, 18 grams of the radioactive element sodium-24 decayed to 6 grams in 24 hours. Find its half-life to the nearest hour. Solution So

22. 22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Finding the Half-Life of a Substance Solution continued Use the formula

23. 23 © 2010 Pearson Education, Inc. All rights reserved RADIOCARBON DATING Carbon-14 ( 14 C) is a form of carbon that decays radioactively with a half-life (time required for half of any given mass to decay) of 5700 years. After the organism dies, the age of its remains can be calculated by determining how much carbon-14 has decayed.

24. 24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 King Tut’s Treasure In 1960, a group of specialists investigated whether a piece of art containing organic material found in Tutankhamun’s tomb had been made during his reign or 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?

25. 25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 King Tut’s Treasure Solution The half-life of carbon-14 is approximately 5700 years, so we rewrite A(t) = A 0 e kt as Now find k.

26. 26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 King Tut’s Treasure Solution continued Substituting this value into the equation yields,

27. 27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 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 of the original amount of carbon- 14 remaining in the object (after 3306 years) is 66.897%.

28. 28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 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. 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%.