College Algebra Twelfth Edition Chapter 4 Inverse Exponential
College Algebra Twelfth Edition Chapter 4 Inverse, Exponential, and Logarithmic Functions Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 1
Section 4. 5 Exponential and Logarithmic Equations • Exponential Equations • Logarithmic Equations • Applications and Models Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 2
Property of Logarithms If x > 0, y > 0, and following holds. x = y is equivalent to then the Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3
Example 1: Solving an Exponential Equation (1 of 2) Give the solution to the nearest Solve thousandth. Solution The properties of exponents cannot be used to solve this equation, so we apply the preceding property of logarithms. While any appropriate base b can be used, the best practical base is base 10 or base e. We choose base e (natural) logarithms here. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4
Example 1: Solving an Exponential Equation (2 of 2) Give the solution to the nearest Solve thousandth. Solution Property of logarithms Power property Divide by Use a calculator. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5
Caution (1 of 2) When evaluating a quotient like in Example 1, do not confuse this quotient with which can be written as We cannot change the quotient of two logarithms to a difference of logarithms. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6
Example 2: Solving an Exponential Equation (1 of 3) Solve thousandth. Give the solution to the nearest Solution Take the natural logarithm on each side. Power property Distributive property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7
Example 2: Solving an Exponential Equation (2 of 3) Solve thousandth. Solution Give the solution to the nearest Write the terms with x on one side Factor out x. Divide by Power property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8
Example 2: Solving an Exponential Equation (3 of 3) Solve thousandth. Give the solution to the nearest Solution Apply the exponents. Product and quotient properties. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9
Example 3: Solving Base e Exponential Equations (1 of 4) Solve each equation. Give solutions to the nearest thousandth. (a) Solution Take the natural logarithm on each side. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10
Example 3: Solving Base e Exponential Equations (2 of 4) Solve each equation. Give solutions to the nearest thousandth. (a) Solution Square root property Use a calculator. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11
Example 3: Solving Base e Exponential Equations (3 of 4) Solve each equation. Give solutions to the nearest thousandth. (b) Solution Divide by e; Take the natural logarithm on each side. Power property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12
Example 3: Solving Base e Exponential Equations (4 of 4) Solve each equation. Give solutions to the nearest thousandth. (b) Solution Multiply by Use a calculator. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13
Example 4: Solving an Exponential Equation (Quadratic in Form) (1 of 2) Solve Give exact value(s) for x. Solution If we substitute we notice that the equation is quadratic in form. Let Factor. Zero-factor property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14
Example 4: Solving an Exponential Equation (Quadratic in Form) (2 of 2) Solve Give exact value(s) for x. Solution Solve for u. Substitute for u. Take the natural logarithm on each side. Both values check, so the solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15
Example 5: Solving Logarithmic Equations (1 of 2) Solve each equation. Give exact values. (a) Solution Divide by 7. Write the natural logarithm in exponential form. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16
Example 5: Solving Logarithmic Equations (2 of 2) Solve each equation. Give exact values. (b) Solution Write in exponential form. Apply the exponent. Add 19. Take cube roots. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 17
Example 6: Solving a Logarithmic Equation (1 of 2) Give exact value(s). Solve Solution Keep in mind that logarithms are defined only for nonnegative numbers. Quotient property Property of logarithms Multiply by x + 2. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 18
Example 6: Solving a Logarithmic Equation (2 of 2) Solve Give exact value(s). Solution Distributive property Standard form Factor. Zero-factor property Solve for x. The proposed negative solution (− 3) is not in the domain in the original equation, so the only valid solution of is the positive number 2. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 19
Caution (2 of 2) Recall that the domain of For this reason, it is always necessary to check that proposed solutions of a logarithmic equation result in logarithms of positive numbers in the original equation. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20
Example 7: Solving a Logarithmic Equation (1 of 2) Solve Give exact value(s). Solution Write in exponential form. Multiply. Standard form Factor. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21
Example 7: Solving a Logarithmic Equation (2 of 2) Give exact value(s). Solve Solution Zero-factor property Solve for x. A check is necessary to be sure that the argument of the logarithm in the given equation is positive. In both cases, the product is true. leads to 8, and The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 22
Example 8: Solving a Logarithmic Equation (1 of 4) Solve Give exact value(s). Solution Product property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 23
Example 8: Solving a Logarithmic Equation (2 of 4) Solve Give exact value(s). Solution Multiply. Subtract 10. Quadratic formula Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 24
Example 8: Solving a Logarithmic Equation (3 of 4) Solve Give exact value(s). Solution The two proposed solutions are The first of these proposed solutions, is negative and when substituted for x in results in a negative argument, which is not allowed. Therefore, this solution must be rejected. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 25
Example 8: Solving a Logarithmic Equation (4 of 4) Solve Give exact value(s). Solution The two proposed solutions are The second proposed solution, is positive. Substituting it for x in results in a positive argument, and substituting it for x in also results in a positive argument, both of which are necessary conditions. Therefore, the solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 26
Note We could have used the definition of logarithm in Example 8 by first writing Equation from Example 8 Substitute Definition of logarithm and then continuing as shown in the solution. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 27
Example 9: Solving a Base e Logarithmic Equation (1 of 2) Solve Solution Give exact value(s). Quotient property Property of logarithms Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 28
Example 9: Solving a Base e Logarithmic Equation (2 of 2) Solve Solution Give exact value(s). Multiply by x − 3. Distributive property Solve for x. Check that the solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 29
Solving Exponential or Logarithmic Equations (1 of 2) To solve an exponential or logarithmic equation, change the given equation into one of the following forms, where a and b are real numbers, a > 0 and follow the guidelines. 1. Solve by taking logarithms on both sides. 2. Solve by changing to exponential form 3. The given equation is equivalent to the equation Solve algebraically. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 30
Solving Exponential or Logarithmic Equations (2 of 2) 4. In a more complicated equation, such as in Example 3(b), it may be necessary to first solve for and then solve the resulting equation using one of the methods given above. 5. Check that each proposed solution is in the domain. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 31
Example 10: Applying an Exponential Equation to the Strength of a Habit (1 of 4) The strength of a habit is a function of the number of times the habit is repeated. If N is the number of repetitions and H is the strength of the habit, then, according to psychologist C. L. Hull, where k is a constant. Solve this equation for k. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 32
Example 10: Applying an Exponential Equation to the Strength of a Habit (2 of 4) Solution First solve the equation for Divide by 1000. Subtract 1. Multiply by − 1 and rewrite Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 33
Example 10: Applying an Exponential Equation to the Strength of a Habit (3 of 4) Solution Now solve for k. Take the natural logarithm on each side. Multiply by Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 34
Example 10: Applying an Exponential Equation to the Strength of a Habit (4 of 4) Solution With the final equation, if one pair of values for H and N is known, k can be found, and the equation can then be used to find either H or N for given values of the other variable. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 35
Example 11: Modeling PC Tablet Sales in the U. S. (1 of 4) The table gives U. S. tablet sales (in millions) for several years. The data can be modeled by the function Year Sales (in millions) 2010 10. 3 2011 24. 1 2012 35. 1 2013 39. 8 2014 42. 1 where t is the number of years after 2009. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 36
Example 11: Modeling PC Tablet Sales in the U. S. (2 of 4) (a) Use the function to estimate the number of tablets sold in the United States in 2015? Solution The year 2015 is represented by t = 2015 − 2009 = 6. Let t = 6. Use a calculator. Based on this model, 47. 4 million tablets were sold in 2015. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 37
Example 11: Modeling PC Tablet Sales in the U. S. (3 of 4) (b) If this trend continues, approximately when will annual sales reach 60 million? Solution Replace with 60, and solve for t. Subtract 10. 58. Divide by 20. 57 and rewrite. Write in exponential form. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 38
Example 11: Modeling PC Tablet Sales in the U. S. (4 of 4) (b) If this trend continues, approximately when will annual sales reach 60 million? Solution Use a calculator. Adding 11 to 2009 gives the year 2020. Based on this model, annual sales will reach 60 million in 2020. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 39
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