7 5 Exponential and Logarithmic Equations and Inequalities

7 -5 Exponential and Logarithmic Equations and Inequalities Objectives Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations. Holt Algebra 2

7 -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. Helpful Hint When you use a rounded number in a check, the result will not be exact, but it should be reasonable. Holt Algebra 2

7 -5 Exponential and Logarithmic Equations and Inequalities Solve and check. 98 – x = 27 x – 3 (32)8 – x = (33)x – 3 316 – 2 x = 33 x – 9 16 – 2 x = 3 x – 9 x=5 Holt Algebra 2 Rewrite each side with the same base; 9 and 27 are powers of 3. To raise a power to a power, multiply exponents. Bases are the same, so the exponents must be equal. Solve for x.

7 -5 Exponential and Logarithmic Equations and Inequalities Check 98 – x = 27 x – 3 98 – 5 275 – 3 93 272 729 The solution is x = 5. Holt Algebra 2

7 -5 Exponential and Logarithmic Equations and Inequalities Solve and check. 4 x – 1 = 5 log 4 x – 1 = log 5 (x – 1)log 4 = log 5 x – 1 = log 5 log 4 5 is not a power of 4, so take the log of both sides. Apply the Power Property of Logarithms. Divide both sides by log 4. log 5 x = 1 + log 4 ≈ 2. 161 The solution is x ≈ 2. 161. Holt Algebra 2 Check Use a calculator.

7 -5 Exponential and Logarithmic Equations and Inequalities 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 Solve for x. Check 32 x = 27 32(1. 5) 27 33 27 27 27 Holt Algebra 2

7 -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 Algebra 2 Check

Exponential and Logarithmic 7 -5 Equations and Inequalities 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 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. Check

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. Remember! Review the properties of logarithms from Lesson 7 -4. Holt Algebra 2

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

7 -5 Exponential and Logarithmic Equations and Inequalities Solve. log 4100 – log 4(x + 1) = 1 log 4(x + 1 ) = 1 100 4 100 log 4( x + 1 ) 100 x+1 = 41 Use 4 as the base for both sides. =4 Use inverse properties on the left side. x = 24 Holt Algebra 2 Write as a quotient.

7 -5 Exponential and Logarithmic Equations and Inequalities Solve. log 5 x 4 = 8 4 log 5 x = 8 log 5 x = 2 x = 52 x = 25 Holt Algebra 2 Power Property of Logarithms. Divide both sides by 4 to isolate log 5 x. Definition of a logarithm.

7 -5 Exponential and Logarithmic Equations and Inequalities 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 Algebra 2 Product Property of Logarithms. Exponential form. Use the inverse properties.

7 -5 Exponential and Logarithmic Equations and Inequalities 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 Algebra 2 1 1 log 12( – 4) + log 12(– 4 +1) 1 x log 12( – 4) is undefined. The solution is x = 3.

7 -5 Exponential and Logarithmic Equations and Inequalities 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 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.

7 -5 Exponential and Logarithmic Equations and Inequalities 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 Algebra 2 Use inverse properties on the left side.

7 -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 Algebra 2