1. Section 6 Chapter 10

2. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 3 4 Exponential and Logarithmic Equations; Further Applications Solve equations involving variables in the exponents. Solve equations involving logarithms. Solve applications of compound interest. Solve applications involving base e exponential growth and decay. 10.6

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Properties for Solving Exponential and Logarithmic Equations For all real numbers b > 0, b ≠ 1, and any real numbers x and y, the following are true. 1. If x = y, then b x = b y. 2. If b x = b y, then x = y. 3. If x = y, and x > 0, y > 0, then log b x = log b y. 4. If x > 0, y > 0, and log b x = log b y, then x = y. Slide 10.6- 3 Exponential and Logarithmic Equations; Further Applications

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve equations involving variables in the exponents. Objective 1 Slide 10.6- 4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve the equation. Approximate the solution to three decimal places. 10 x = 4 log 10 x = log 4 x log 10 = log 4 x ≈ 0.602 Check. x = 0.602 10 0.6021 ≈ 4 The solution set is {0.602}. Slide 10.6- 5 CLASSROOM EXAMPLE 1 Solving an Exponential Equation Solution:

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve e 0.02x = 192. Approximate the solution to three decimal places. Check. x = 262.875 The solution set is {262.875}. ln e 0.02x = ln 192 0.02x ln e = ln 192 0.02x = ln 192 e 0.02(262.875) ≈ 192 Slide 10.6- 6 CLASSROOM EXAMPLE 2 Solving an Exponential Equation with Base e Solution:

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. General Method for Solving an Exponential Equation Take logarithms to the same base on both sides and then use the power rule of logarithms or the special property log b b x = x. (See Examples 1 and 2.) As a special case, if both sides can be written as exponentials with the same base, do so, and set the exponents equal. (See Section 10.2.) Slide 10.6- 7 Solve equations involving variables in the exponents.

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve equations involving logarithms. Objective 2 Slide 10.6- 8

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve log 3 (x + 1) 5 = 3. Give the exact solution. Check. log 3 (x + 1) 5 = 3 (x + 1) 5 = 3 3 (x + 1) 5 = 27 x + 1 = x = –1 + log 3 The solution set is { }. Slide 10.6- 9 CLASSROOM EXAMPLE 3 Solving a Logarithmic Equation Solution: True

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve log 8 (2x + 5) + log 8 3 = log 8 33. Check. The solution set is {3}. x = 3 log 8 11 + log 8 3 = log 8 33 log 8 (2x + 5) + log 8 3 = log 8 33 log 8 [3(2x + 5)] = log 8 33 3(2x + 5) = 33 2x + 5 = 11 2x = 6 x = 3 True Slide 10.6- 10 CLASSROOM EXAMPLE 4 Solving a Logarithmic Equation Solution:

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve log x + log (x + 15) = 2 The value − 25 must be rejected as a solution since it leads to the logarithm of a negative number in the original equation. The solution set is {5}. Slide 10.6- 11 CLASSROOM EXAMPLE 5 Solving a Logarithmic Equation Solution:

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving a Logarithmic Equation Step 1 Transform the equation so that a single logarithm appears on one side. Use the product rule or quotient rule of logarithms to do this. Step 2 (a) Use Property 4. If log b x = log b y, then x = y. (See Example 4.) (b) Write the equation in exponential form. If log b x = k, then x = b k. (See Examples 3 and 5). Slide 10.6- 12 Solve equations involving logarithms.

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve applications of compound interest. Objective 3 Slide 10.6- 13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Compound Interest Formula (for a Finite Number of Periods) If a principal of P dollars is deposited at an annual rate of interest r compounded (paid) n times per year, then the account will contain dollars after t years. (In this formula, r is expressed as a decimal.) Slide 10.6- 14 Solve applications of compound interest.

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the value of $2000 deposited at 4% compounded quarterly for 10 years. Use with P = 2000, r = 4% = 0.04, n = 4 (compounded quarterly), and t = 10. The value is about $2977.73 Slide 10.6- 15 CLASSROOM EXAMPLE 6 Solving a Compound Interest Problem for A Solution:

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Approximate the time it would take for money paying 4% interest compounded semiannually to double. Round to the nearest hundredth. Divide by P. Semiannually (n = 2). Slide 10.6- 16 CLASSROOM EXAMPLE 7 Solving a Compound Interest Problem for t Solution: Property 3 Power rule

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Divide by log 1.02. Divide by 2. It will take about 17.50 yr for money paying 4% interest compounded semiannually to double. Slide 10.6- 17 CLASSROOM EXAMPLE 7 Solving a Compound Interest Problem for t (cont’d)

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Continuous Compound Interest Formula If a principal of P dollars is deposited at an annual rate of interest r compounded continuously for t years, the final amount on deposit is Slide 10.6- 18 Solve applications of compound interest.

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Suppose that $2000 is invested at 5% interest for 10 yr. How much will the investment grow to if it is compounded continuously? It will grow to $3297.44. Slide 10.6- 19 CLASSROOM EXAMPLE 8 Solving a Continuous Compound Interest Problem Solution:

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. How long will it take for the amount to double? It would take about 13.9 yr for the initial investment to double. Slide 10.6- 20 CLASSROOM EXAMPLE 8 Solving a Continuous Compound Interest Problem (cont’d) Solution:

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve applications involving base e exponential growth and decay. Objective 4 Slide 10.6- 21

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Radioactive strontium decays according to the function defined by y = y 0 e -0.0239t, where t is time in years and y 0 is the amount present at time t = 0. If an initial sample contains y 0 = 12 g of radioactive strontium, how many grams will be present after 35 yr? After 35 yr, there will be about 5.20 grams of radioactive strontium. Slide 10.6- 22 CLASSROOM EXAMPLE 9 Solving an Application Involving Exponential Decay Solution:

23. Copyright © 2012, 2008, 2004 Pearson Education, Inc. What is the half-life of radioactive strontium? The half-life is about 29 years. Slide 10.6- 23 CLASSROOM EXAMPLE 9 Solving an Application Involving Exponential Decay (cont’d) Solution: