1. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities 4-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

2. 4-5 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 4-5 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.

4. Holt McDougal Algebra 2 4-5 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

5. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 9 8 – x = 27 x – 3 (3 2 ) 8 – x = (3 3 ) x – 3 Rewrite each side with the same base; 9 and 27 are powers of 3. 3 16 – 2x = 3 3x – 9 To raise a power to a power, multiply exponents. Example 1A: Solving Exponential Equations 16 – 2x = 3x – 9 Bases are the same, so the exponents must be equal. x = 5 Solve for x.

6. Holt McDougal Algebra 2 4-5 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 1B: Solving Exponential Equations 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

7. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 3 2x = 27 (3) 2x = (3) 3 Rewrite each side with the same base; 3 and 27 are powers of 3. 3 2x = 3 3 To raise a power to a power, multiply exponents. Check It Out! Example 1a 2x = 3 Bases are the same, so the exponents must be equal. x = 1.5 Solve for x.

8. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 7 –x = 21 log 7 –x = log 21 21 is not a power of 7, so take the log of both sides. (–x)log 7 = log 21 Apply the Power Property of Logarithms. Check It Out! Example 1b Divide both sides by log 7. x = – ≈ –1.565 log21 log7 – x = log21 log7

9. Holt McDougal Algebra 2 4-5 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. Check It Out! Example 1c Divide both sides by log 2, then divide both sides by 3. x ≈ 1.302 3x = log15 log2

10. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3A: Solving Logarithmic Equations Use 6 as the base for both sides. log 6 (2x – 1) = –1 6 log 6 (2x –1) = 6 –1 2x – 1 = 1 6 7 12 x =x = Use inverse properties to remove 6 to the log base 6. Simplify.

11. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3B: Solving Logarithmic Equations Write as a quotient. log 4 100 – log 4 (x + 1) = 1 x = 24 Use 4 as the base for both sides. Use inverse properties on the left side. 100 x + 1 log 4 ( ) = 1 4 log 4 = 4 1 100 x + 1 ( ) = 4 100 x + 1

12. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3C: 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.

13. Holt McDougal Algebra 2 4-5 Exponential and Logarithmic Equations and Inequalities Solve. Example 3D: Solving Logarithmic Equations Product Property of Logarithms. log 12 x + log 12 (x + 1) = 1 Exponential form. Use the inverse properties. log 12 x(x + 1) = 1 log 12 x(x +1) 12 = 12 1 x(x + 1) = 12

14. Holt McDougal Algebra 2 4-5 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.

15. Holt McDougal Algebra 2 4-5 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.

16. Holt McDougal Algebra 2 4-5 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

17. Holt McDougal Algebra 2 4-5 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}.

18. Holt McDougal Algebra 2 4-5 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