1. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. 1. log 16 x = 2. log x 1.331 = 3 3. log10,000 = x 3 2

2. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations. Objectives

3. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities exponential equation logarithmic equation Vocabulary

4. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations: Try writing them so that the bases are all the same. Take the logarithm of both sides.

5. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities When you use a rounded number in a check, the result will not be exact, but it should be reasonable. Helpful Hint

6. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve and check. 9 8 – x = 27 x – 3 Example 1A: Solving Exponential Equations x = 5 Solve for x.

7. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve and check. 4 x – 1 = 5 Example 1B: Solving Exponential Equations The solution is x ≈ 2.161.

8. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve and check. 3 2x = 27 Check It Out! Example 1a

9. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve and check. 7 –x = 21 Check It Out! Example 1b

10. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve and check. 2 3x = 15 Check It Out! Example 1c

11. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000? Example 2: Biology Application

12. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities You receive one penny on the first day, and then triple that (3 cents) on the second day, and so on for a month. On what day would you receive a least a million dollars. Check It Out! Example 2

13. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.

14. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Review the properties of logarithms from Lesson 7-4. Remember!

15. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. Example 3A: Solving Logarithmic Equations log 6 (2x – 1) = –1

16. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. Example 3B: Solving Logarithmic Equations log 4 100 – log 4 (x + 1) = 1

17. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. Example 3C: Solving Logarithmic Equations log 5 x 4 = 8

18. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. Example 3D: Solving Logarithmic Equations log 12 x + log 12 (x + 1) = 1

19. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. 3 = log 8 + 3log x Check It Out! Example 3a

20. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Solve. 2log x – log 4 = 0 Check It Out! Example 3b

21. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Watch out for calculated solutions that are not solutions of the original equation. Caution

22. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Use a table and graph to solve 2 x + 1 > 8192x. Example 4A: Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities Use a graphing calculator. Enter 2^(x + 1) as Y1 and 8192x as Y2.

23. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Use a table and graph to solve 2 x + 1 > 8192x. Example 4A: Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities Use a graphing calculator. Enter 2^(x + 1) as Y1 and 8192x as Y2. In the table, find the x-values where Y1 is greater than Y2. In the graph, find the x-value at the point of intersection. The solution set is {x | x > 16}.

24. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities log(x + 70) = 2log ( ) x 3 Use a graphing calculator. Enter log(x + 70) as Y1 and 2log( ) as Y2. x 3 Example 4B

25. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities log(x + 70) = 2log ( ) In the table, find the x-values where Y1 equals Y2. In the graph, find the x-value at the point of intersection. x 3 Use a graphing calculator. Enter log(x + 70) as Y1 and 2log( ) as Y2. x 3 The solution is x = 30. Example 4B

26. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 4a Use a table and graph to solve 2 x = 4 x – 1. Use a graphing calculator. Enter 2 x as Y1 and 4 (x – 1) as Y2.

27. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities In the table, find the x-values where Y1 is equal to Y2. In the graph, find the x-value at the point of intersection. Check It Out! Example 4a Use a table and graph to solve 2 x = 4 x – 1. Use a graphing calculator. Enter 2 x as Y1 and 4 (x – 1) as Y2. The solution is x = 2.

28. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 4b Use a table and graph to solve 2 x > 4 x – 1. Use a graphing calculator. Enter 2 x as Y1 and 4 (x – 1) as Y2.

29. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities In the table, find the x-values where Y1 is greater than Y2. In the graph, find the x-value at the point of intersection. Check It Out! Example 4b Use a table and graph to solve 2 x > 4 x – 1. Use a graphing calculator. Enter 2 x as Y1 and 4 (x – 1) as Y2. The solution is x < 2.

30. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 4c Use a table and graph to solve log x 2 = 6.

31. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities In the table, find the x-values where Y1 is equal to Y2. In the graph, find the x-value at the point of intersection. Check It Out! Example 4c Use a table and graph to solve log x 2 = 6. Use a graphing calculator. Enter log(x 2 ) as Y1 and 6 as Y2. The solution is x = 1000.

32. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Lesson Quiz: Part I Solve. 1. 4 3x–1 = 8 x+1 2. 3 2x–1 = 20 3. log 7 (5x + 3) = 3 4. log(3x + 1) – log 4 = 2 5. log 4 (x – 1) + log 4 (3x – 1) = 2

33. Holt McDougal Algebra 2 Exponential and Logarithmic Equations and Inequalities Lesson Quiz: Part II 6. A single cell divides every 5 minutes. How long will it take for one cell to become more than 10,000 cells? 7. Use a table and graph to solve the equation 2 3x = 3 3x–1.