8-5 Exponential and Logarithmic Equations.pdf

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 20098-5 **Exponential** & **Logarithmic****Equations**Objectives:• Solve exponential equations.• Solve logarithmic equations.Mar 202:11 PM1

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Check Skills You'll NeedEvaluate each logarithm.1. log 9 81 log 9 3 2. log 10 log 3 93. log 2 16 ÷ log 2 8 4. Simplify 125 2 3Mar 202:13 PM2

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Solving **Exponential** **Equations**An equation of the form b cx =a, where the exponentincludes a variable, is an exponential equation.If m **and** n are positive **and** m = n, then log m = log n.Therefore, you can solve an exponential equation bytaking the logarithm of each side of the equation.Mar 202:14 PM3

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #1: Solving an **Exponential** EquationSolve 7 3x = 20.7 3x = 20log 7 3x = log 203x log 7 = log 20x =log 203log 7x ≈ 0.5132Check: 7 3x = 207 3(0.5132) = 2020.00382 ≈ 20Take the common logarithm of each side.Use the power property of logarithms.Divide each side by 3 log 7.Use a calculator.Mar 202:14 PM4

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #2: Solve each equation. Round to thenearest ten-thous**and**th. Check your answers.a. 3 x = 4 b. 6 2x = 21 c. 3 x+4 = 101Mar 202:15 PM5

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Solving **Logarithmic** **Equations**To evaluate a logarithm with any base, you can use theChange of Base Formula.Mar 202:17 PM6

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #3: Using the Change of Base FormulaUse the Change of Base Formula to evaluate log 3 15.log 3 15 =log 15log 3≈ 2.4650Mar 202:18 PM7

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #4: Evaluate log 5 400.Mar 202:18 PM8

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009An equation that includes a logarithmicexpression, such as log 3 15 = log 2 x is called alogarithmic equation.Mar 202:18 PM9

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #5: Solving a **Logarithmic** EquationSolve log (3x + 1) = 5.log (3x + 1) = 53x + 1 = 10 53x + 1 = 100,0003x = 99,999x = 33,333Check: log (3x + 1) = 5log (3(33,333) + 1) = 5log (100,000) = 5log 10 5 = 55 = 5Mar 202:18 PM10

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #6: Solve log (7 2x) = 1. Check your answer.Mar 202:18 PM11

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #7: Using **Logarithmic** Properties to Solve an EqauationSolve 2 log x log 3 = 2.2 log x log 3 = 2(x 2)log = 2 Write as a single logarithm.3x 23= 10 2 Write in exponential form.x 2 = 3(100)x = ±10√3 ≈ ±17.32Log x is defined only for x>0, so the solution is 10√3 or about 17.32.Mar 202:19 PM12

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Example #8: Solve log 6 log 3x = 2.Mar 202:19 PM13

85 **Exponential** **and** **Logarithmic** **Equations**April 08, 2009Homework: page 464(1 - 12, 23, 25 - 32 evaluate, 33 - 45)Mar 202:19 PM14