1. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve logarithmic equations. Objectives

2. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 4 x – 1 = 5 log 4 x – 1 = log 5 5 is not a power of 4, so take the log of both sides. (x – 1)log 4 = log 5 Apply the Power Property of Logarithms. Example 1 Divide both sides by log 4. Check Use a calculator. The solution is x ≈ 2.161. x = 1 + ≈ 2.161 log5 log4 x –1 = log5 log4

3. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 2 3x = 15 log2 3x = log15 15 is not a power of 2, so take the log of both sides. (3x)log 2 = log15 Apply the Power Property of Logarithms. Example 1b Divide both sides by log 2, then divide both sides by 3. x ≈ 1.302 3x = log15 log2

4. Holt McDougal Algebra 2 7-5 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.

5. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 2: Solving Logarithmic Equations log 6 (2x – 1) = –1

6. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3: Solving Logarithmic Equations Write as a quotient. log 4 100 – log 4 (x + 1) = 1 100 x + 1 log 4 ( ) = 1

7. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3B: Solving Logarithmic Equations Power Property of Logarithms. log 5 x 4 = 8 x = 25 Definition of a logarithm. 4log 5 x = 8 log 5 x = 2 x = 5 2 Divide both sides by 4 to isolate log 5 x.

8. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3C: Solving Logarithmic Equations Product Property of Logarithms. log 12 x + log 12 (x + 1) = 1 log 12 x(x + 1) = 1

9. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Example 3 Continued Multiply and collect terms. Factor. Solve. x 2 + x – 12 = 0 log 12 x + log 12 (x +1) = 1 (x – 3)(x + 4) = 0 x – 3 = 0 or x + 4 = 0 Set each of the factors equal to zero. x = 3 or x = –4 log 12 x + log 12 (x +1) = 1 log 12 3 + log 12 (3 + 1) 1 log 12 3 + log 12 4 1 log 12 12 1 The solution is x = 3. 1 log 12 ( –4) + log 12 (–4 +1) 1 log 12 ( –4) is undefined. x Check Check both solutions in the original equation.

10. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. 3 = log 8 + 3log x Check It Out! Example 3a 3 = log 8 + 3log x 3 = log 8 + log x 3 3 = log (8x 3 ) 10 3 = 10 log (8x 3 ) 1000 = 8x 3 125 = x 3 5 = x Use 10 as the base for both sides. Use inverse properties on the right side. Product Property of Logarithms. Power Property of Logarithms.

11. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. 2log x – log 4 = 0 Check It Out! Example 3b Write as a quotient. x = 2 Use 10 as the base for both sides. Use inverse properties on the left side. 2log ( ) = 0 x 4 2(10 log ) = 10 0 x 4 2( ) = 1 x 4

12. Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Watch out for calculated solutions that are not solutions of the original equation. Caution

13. Holt McDougal Algebra 2 7-5 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 4 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.

14. Holt McDougal Algebra 2 7-5 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 x ≈ 1.86 x = 68 x = 133 x = 3 x = 5 3