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 = 27x – 3 (32)8 – x = (33)x – 3 316 – 2x = 33x – 9 16 – 2x = 3x – 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 = 27x – 3 98 – 5 275 – 3 93 272 729 729 The solution is x = 5. Holt Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 4x – 1 = 5 log 4x – 1 = log 5 (x – 1)log 4 = log 5 log5 x –1 = log4 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. log5 x = 1 + log4 ≈ 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. 32x = 27 2x (3) 3 = (3) Rewrite each side with the same base; 3 and 27 are powers of 3. 32x = 33 To raise a power to a power, multiply exponents. 2x = 3 Bases are the same, so the exponents must be equal. x = 1.5 Solve for x. Check 32x = 27 32(1.5) 27 33 27 27 27 Holt Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Check It Out! Example 1b Solve and check. 7–x = 21 log 7–x = log 21 (–x)log 7 = log 21 log21 –x = log7 log21 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 = – log7 ≈ –1.565 Holt Algebra 2 Check Exponential and Logarithmic 7-5 Equations and Inequalities Solve and check. 23x = 15 log23x = log15 (3x)log 2 = log15 log15 3x = log2 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. log6(2x – 1) = –1 6 log (2x –1) 6 = 6–1 2x – 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. log4100 – log4(x + 1) = 1 100 log4(x + 1 ) = 1 log4( x + 1 ) 100 4 = 41 100 =4 x+1 x = 24 Holt Algebra 2 Write as a quotient. Use 4 as the base for both sides. Use inverse properties on the left side. 7-5 Exponential and Logarithmic Equations and Inequalities Solve. log5x 4 = 8 4log5x = 8 log5x = 2 x = 52 x = 25 Holt Algebra 2 Power Property of Logarithms. Divide both sides by 4 to isolate log5x. Definition of a logarithm. 7-5 Exponential and Logarithmic Equations and Inequalities Solve. log12x + log12(x + 1) = 1 log12 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 x2 + 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 Check both solutions in the original equation. log12x + log12(x +1) = 1 log12x + log12(x +1) = 1 log123 + log12(3 + 1) log123 + log124 log1212 1 1 1 1 1 log12( –4) + log12(–4 +1) 1 x log12( –4) is undefined. The solution is x = 3. Holt Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities Solve. 3 = log 8 + 3log x 3 = log 8 + 3log x 3 = log 8 + log x3 3 = log (8x3) 103 = 10log (8x3) 1000 = 8x3 125 = x3 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. 2log x – log 4 = 0 x 2log( 4 log 2(10 )=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. 43x–1 = 8x+1 x= 5 3 2. 32x–1 = 20 x ≈ 1.86 3. log7(5x + 3) = 3 x = 68 4. log(3x + 1) – log 4 = 2 x = 133 5. log4(x – 1) + log4(3x – 1) = 2 x=3 Holt Algebra 2