4 5 Exponential and Logarithmic Equations and Inequalities

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

4 -5 Exponential and Logarithmic Equations and Inequalities Warm Up Solve. 1. log 16 x = 3 2 64 2. logx 1. 331 = 3 1. 1 3. log 10, 000 = x 4 Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Objectives Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Vocabulary exponential equation logarithmic equation Holt Mc. Dougal 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. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Helpful Hint When you use a rounded number in a check, the result will not be exact, but it should be reasonable. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 1 A: Solving Exponential Equations Solve and check. 98 – x = 27 x – 3 (32)8 – x = (33)x – 3 Rewrite each side with the same base; 9 and 27 are powers of 3. To raise a power to a power, 316 – 2 x = 33 x – 9 multiply exponents. 16 – 2 x = 3 x – 9 Bases are the same, so the exponents must be equal. x=5 Holt Mc. Dougal Algebra 2 Solve for x.

4 -5 Exponential and Logarithmic Equations and Inequalities Example 1 A Continued Check 98 – x = 27 x – 3 98 – 5 275 – 3 93 272 729 The solution is x = 5. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 1 B: Solving Exponential Equations Solve and check. 4 x – 1 = 5 5 is not a power of 4, so take the log 4 x – 1 = log 5 log of both sides. Apply the Power Property of (x – 1)log 4 = log 5 Logarithms. log 5 x – 1 = log 4 Divide both sides by log 4. log 5 x = 1 + log 4 ≈ 2. 161 Check Use a calculator. The solution is x ≈ 2. 161. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1 a Solve and check. 32 x = 27 (3)2 x = (3)3 Rewrite each side with the same base; 3 and 27 are powers of 3. 32 x = 33 To raise a power to a power, multiply exponents. 2 x = 3 Bases are the same, so the exponents must be equal. x = 1. 5 Holt Mc. Dougal Algebra 2 Solve for x.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1 a Continued Check 32 x = 27 32(1. 5) 27 33 27 27 27 The solution is x = 1. 5. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1 b Solve and check. 7–x = 21 log 7–x = log 21 (–x)log 7 = log 21 –x = log 21 log 7 log 21 21 is not a power of 7, so take the log of both sides. Apply the Power Property of Logarithms. Divide both sides by log 7. x = – log 7 ≈ – 1. 565 Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1 b Continued Check Use a calculator. The solution is x ≈ – 1. 565. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1 c Solve and check. 23 x = 15 log 23 x = log 15 (3 x)log 2 = log 15 3 x = log 15 log 2 x ≈ 1. 302 Holt Mc. Dougal Algebra 2 15 is not a power of 2, so take the log of both sides. Apply the Power Property of Logarithms. Divide both sides by log 2, then divide both sides by 3.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1 c Continued Check Use a calculator. The solution is x ≈ 1. 302. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 2: Biology Application Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1, 000? At hour 0, there is one bacterium, or 20 bacteria. At hour one, there are two bacteria, or 21 bacteria, and so on. So, at hour n there will be 2 n bacteria. Solve 2 n > 106 Write 1, 000 in scientific annotation. log 2 n > log 106 Take the log of both sides. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 2 Continued nlog 2 > log 106 Use the Power of Logarithms. nlog 2 > 6 log 106 is 6. 6 n > log 2 n> 6 0. 301 n > ≈ 19. 94 Divide both sides by log 2. Evaluate by using a calculator. Round up to the next whole number. It will take about 20 hours for the number of bacteria to exceed 1, 000. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 2 Continued Check In 20 hours, there will be 220 bacteria. 220 = 1, 048, 576 bacteria. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 2 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. $1, 000 is 100, 000 cents. On day 1, you would receive 1 cent or 30 cents. On day 2, you would receive 3 cents or 31 cents, and so on. So, on day n you would receive 3 n– 1 cents. Solve 3 n – 1 > 1 x 108 Write 100, 000 in scientific annotation. log 3 n – 1 > log 108 Take the log of both sides. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 2 Continued (n – 1) log 3 > log 108 Use the Power of Logarithms. (n – 1)log 3 > 8 8 n – 1 > log 3 8 log 108 is 8. Divide both sides by log 3. n > log 3 + 1 Evaluate by using a calculator. n > ≈ 17. 8 Round up to the next whole number. Beginning on day 18, you would receive more than a million dollars. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 2 Check On day 18, you would receive 318 – 1 or over a million dollars. 317 = 129, 140, 163 cents or 1. 29 million dollars. Holt Mc. Dougal Algebra 2

4 -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. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Remember! Review the properties of logarithms from Lesson 7 -4. Holt Mc. Dougal Algebra 2

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

Exponential and Logarithmic Equations and Inequalities 4 -5 Example 3 B: Solving Logarithmic Equations Solve. log 4100 – log 4(x + 1) = 1 log 4(x + 1 ) = 1 100 4 100 log 4( x + 1 ) 100 x+1 Write as a quotient. = 41 Use 4 as the base for both sides. =4 Use inverse properties on the left side. x = 24 Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 3 C: Solving Logarithmic Equations Solve. log 5 x 4 = 8 4 log 5 x = 8 log 5 x = 2 x = 52 x = 25 Holt Mc. Dougal Algebra 2 Power Property of Logarithms. Divide both sides by 4 to isolate log 5 x. Definition of a logarithm.

4 -5 Exponential and Logarithmic Equations and Inequalities Example 3 D: Solving Logarithmic Equations Solve. log 12 x + log 12(x + 1) = 1 log 12 x(x + 1) = 1 12 log x(x +1) 12 = 121 x(x + 1) = 12 Holt Mc. Dougal Algebra 2 Product Property of Logarithms. Exponential form. Use the inverse properties.

4 -5 Exponential and Logarithmic Equations and Inequalities Example 3 Continued x 2 + x – 12 = 0 (x – 3)(x + 4) = 0 Multiply and collect terms. Factor. x – 3 = 0 or x + 4 = 0 Set each of the factors equal to zero. x = 3 or x = – 4 Solve. Check both solutions in the original equation. log 12 x + log 12(x +1) = 1 log 123 + log 12(3 + 1) log 123 + log 124 log 1212 1 Holt Mc. Dougal Algebra 2 1 1 log 12( – 4) + log 12(– 4 +1) 1 x log 12( – 4) is undefined. The solution is x = 3.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 3 a Solve. 3 = log 8 + 3 log x 3 = log 8 + log x 3 3 = log (8 x 3) 103 = 10 log (8 x 3) 1000 = 8 x 3 125 = x 3 5=x Holt Mc. Dougal Algebra 2 Power Property of Logarithms. Product Property of Logarithms. Use 10 as the base for both sides. Use inverse properties on the right side.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 3 b Solve. 2 log x – log 4 = 0 2 log( 4 x 2(10 log )=0 Write as a quotient. x 4 Use 10 as the base for both sides. ) = 100 2( x ) = 1 4 x=2 Holt Mc. Dougal Algebra 2 Use inverse properties on the left side.

4 -5 Exponential and Logarithmic Equations and Inequalities Caution Watch out for calculated solutions that are not solutions of the original equation. Holt Mc. Dougal Algebra 2

Exponential and Logarithmic 4 -5 Equations and Inequalities Example 4 A: Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities Use a table and graph to solve 2 x + 1 > 8192 x. Use a graphing calculator. Enter 2^(x + 1) as Y 1 and 8192 x as Y 2. In the table, find the x-values where Y 1 is greater than Y 2. In the graph, find the x-value at the point of intersection. The solution set is {x | x > 16}. Holt Mc. Dougal Algebra 2

4 -5 Exponential and Logarithmic Equations and Inequalities Example 4 B log(x + 70) = 2 log( x ) 3 Use a graphing calculator. Enter log(x + 70) as Y 1 and 2 log( x ) as Y 2. 3 In the table, find the x-values where Y 1 equals Y 2. The solution is x = 30. Holt Mc. Dougal Algebra 2 In the graph, find the x-value at the point of intersection.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 4 a Use a table and graph to solve 2 x = 4 x – 1. Use a graphing calculator. Enter 2 x as Y 1 and 4(x – 1) as Y 2. In the table, find the x-values where Y 1 is equal to Y 2. The solution is x = 2. Holt Mc. Dougal Algebra 2 In the graph, find the x-value at the point of intersection.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 4 b Use a table and graph to solve 2 x > 4 x – 1. Use a graphing calculator. Enter 2 x as Y 1 and 4(x – 1) as Y 2. In the table, find the x-values where Y 1 is greater than Y 2. The solution is x < 2. Holt Mc. Dougal Algebra 2 In the graph, find the x-value at the point of intersection.

4 -5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 4 c Use a table and graph to solve log x 2 = 6. Use a graphing calculator. Enter log(x 2) as Y 1 and 6 as Y 2. In the table, find the x-values where Y 1 is equal to Y 2. The solution is x = 1000. Holt Mc. Dougal Algebra 2 In the graph, find the x-value at the point of intersection.

4 -5 Exponential and Logarithmic Equations and Inequalities Lesson Quiz: Part I Solve. 1. 43 x– 1 = 8 x+1 x= 5 3 2. 32 x– 1 = 20 x ≈ 1. 86 3. log 7(5 x + 3) = 3 x = 68 4. log(3 x + 1) – log 4 = 2 x = 133 5. log 4(x – 1) + log 4(3 x – 1) = 2 x=3 Holt Mc. Dougal Algebra 2

4 -5 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? 70 min 7. Use a table and graph to solve the equation 23 x = 33 x– 1. x ≈ 0. 903 Holt Mc. Dougal Algebra 2