Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponential and Logarithmic Equations Learn to solve exponential equations. Learn to solve applied problems involving exponential equations. Learn to solve logarithmic equations. Learn how to use the logistic growth model. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solving an Exponential Equation of “Common Base” Type Solve each equation. Solution Solution set is {1}. Solution set is

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Solving Exponential Equations by “Taking Logarithms” Solve each equation and approximate the results to three decimal places. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Solving Exponential Equations by “Taking Logarithms” Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR SOLVING EXPONENTIAL EQUATIONS Step 1.Isolate the exponential expression on one side of the equation. Step 2.Take the common or natural logarithm of both sides of the equation in Step 1. Step 3.Use the power rule log a M r = r log a M to “bring down the exponent.” Step 4.Solve for the variable.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solving an Exponential Equation with Different Bases Solve the equation 5 2x–3 = 3 x+1 and approximate the answer to three decimal places. When different bases are involved begin with Step 2. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 An Exponential Equation of Quadratic Form Solve the equation 3 x – 83 –x = 2. Solution This equation is quadratic in form. Let y = 3 x then y 2 = (3 x ) 2 = 3 2x. Then,

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 An Exponential Equation of Quadratic Form Solution continued But 3 x = –2 is not possible because 3 x > 0 for all numbers x. So, solve 3 x = 4 to find the solution.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 An Exponential Equation of Quadratic Form Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Population Growth Problem The following table shows the approximate population and annual growth rate of the United States and Pakistan in CountryPopulation Annual Population Growth Rate United States295 million1.0% Pakistan162 million3.1%

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Population Growth Problem Use the alternate population model P = P 0 (1 + r) t and the information in the table, and assume that the growth rate for each country stays the same. In this model, P 0 is the initial population and t is the time in years since a.Use the model to estimate the population of each country in b.If the current growth rate continues, in what year will the population of the United States be 350 million?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Population Growth Problem c.If the current growth rate continues, in what year will the population of Pakistan be the same as the population of the United States? Solution Use the given model a.US population in 2005 is P 0 = 295. The year 2015 is 10 years from Pakistan in 2005 is P 0 = 162.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Population Growth Problem Solution continued b.Solve for t to find when the United States population will be 350. Somewhere in the year 2022 ( ) the United States population will be 350.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Population Growth Problem Solution continued c.Solve for t to find when the population will be the same in the two countries.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Population Growth Problem Solution continued Somewhere in the year 2034 ( ) the two populations will be the same.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SOLVING LOGARITHMS EQUATIONS Equations that contain terms of the form log a x are called logarithmic equations. To solve a logarithmic equation we write it in the equivalent exponential form.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solving a Logarithmic Equation Solve: Since the domain of logarithmic functions is positive numbers, we must check our solution. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solving a Logarithmic Equation Solution continued The solution set is Check x = ? ? ? ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using the One-to-One Property of Logarithms Solve: Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using the One-to-One Property of Logarithms Solution continued Check x = 2: ? ? ? ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using the One-to-One Property of Logarithms Solution continued The solution set is {2, 3}. Check x = 3: ? ? ? ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using the Product and Quotient Rules Solve: Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using the Product and Quotient Rules Solution continued Check x = 2: Logarithms are not defined for negative numbers, so x = 2 is not a solution. ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using the Product and Quotient Rules Solution continued The solution set is {5}. Check x = 5: ? ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using the Product and Quotient Rules Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using the Product and Quotient Rules Solution continued The solution set is {4}. Check x = 4: ? ? ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Using the Logistic Growth Model Suppose the carrying capacity M of the human population on Earth is 35 billion. In 1987, the world population was about 5 billion. Use the logistic growth model of P. F. Verhulst to calculate the average rate, k, of growth of the population, given that the population was about 6 billion in 2003.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Using the Logistic Growth Model Solution We have t = 0 (1987), P(t) = 5 and M = 35. We now have

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Using the Logistic Growth Model Solution continued Solve for k given t = 16 (for 2003) and P(t) = 6. The growth rate was approximately 1.35%.