3.4 Exponential and Logarithmic and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic Functions

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  • IntroductionSo far in this chapter, you have studied the definitions, graphs, and properties ofexponential and logarithmic functions. In this section, you will study proceduresfor solving equations involving these exponential and logarithmic functions.

    There are two basic strategies for solving exponential or logarithmicequations. The first is based on the One-to-One Properties and was used to solvesimple exponential and logarithmic equations in Sections 3.1 and 3.2. Thesecond is based on the Inverse Properties. For and the followingproperties are true for all and for which and are defined.

    One-to-One Properties

    if and only if

    if and only if

    Inverse Properties

    Solving Simple Equations

    Original RewrittenEquation Equation Solution Property

    a. One-to-One

    b. One-to-One

    c. One-to-One

    d. Inverse

    e. Inverse

    f. Inverse

    Now try Exercise 13.

    The strategies used in Example 1 are summarized as follows.

    x 101 11010log x 101log x 1

    x e3e ln x e3ln x 3

    x ln 7ln ex ln 7ex 7

    x 23x 3213x

    9

    x 3ln x ln 3ln x ln 3 0

    x 52x 252x 32

    loga ax x

    aloga x x

    x y.loga x loga y

    x y.ax ay

    loga yloga xyxa 1,a > 0

    246 Chapter 3 Exponential and Logarithmic Functions

    What you should learn Solve simple exponential

    and logarithmic equations.

    Solve more complicated exponential equations.

    Solve more complicated logarithmic equations.

    Use exponential and logarith-mic equations to model andsolve real-life problems.

    Why you should learn itExponential and logarithmicequations are used to modeland solve life science applica-tions. For instance, in Exercise112, on page 255, a logarithmicfunction is used to model thenumber of trees per acre giventhe average diameter of thetrees.

    Exponential and Logarithmic Equations

    James Marshall/Corbis

    3.4

    Strategies for Solving Exponential and LogarithmicEquations

    1. Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions.

    2. Rewrite an exponential equation in logarithmic form and apply theInverse Property of logarithmic functions.

    3. Rewrite a logarithmic equation in exponential form and apply the InverseProperty of exponential functions.

    Example 1

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  • Solving Exponential Equations

    Solving Exponential Equations

    Solve each equation and approximate the result to three decimal places if necessary.

    a. b.

    Solutiona. Write original equation.

    One-to-One Property

    Write in general form.

    Factor.

    Set 1st factor equal to 0.

    Set 2nd factor equal to 0.

    The solutions are and Check these in the original equation.

    b. Write original equation.

    Divide each side by 3.

    Take log (base 2) of each side.

    Inverse Property

    Change-of-base formula

    The solution is Check this in the original equation.

    Now try Exercise 25.

    In Example 2(b), the exact solution is and the approximatesolution is An exact answer is preferred when the solution is anintermediate step in a larger problem. For a final answer, an approximate solutionis easier to comprehend.

    Solving an Exponential Equation

    Solve and approximate the result to three decimal places.

    SolutionWrite original equation.

    Subtract 5 from each side.

    Take natural log of each side.

    Inverse Property

    The solution is ln 55 . Check this in the original equation.

    Now try Exercise 51.

    4.007x

    x ln 55 4.007

    ln ex ln 55

    ex 55

    ex 5 60

    ex 5 60

    x 3.807.x log2 14

    x log2 14 3.807.

    x ln 14ln 2

    3.807

    x log2 14

    log2 2x log2 14

    2 x 14

    32 x 42

    x 4.x 1

    x 4 0 x 4 x 1 0 x 1

    x 1x 4 0 x2 3x 4 0

    x2 3x 4

    ex2 e3x4

    32 x 42ex2 e3x4

    Section 3.4 Exponential and Logarithmic Equations 247

    Remember that the natural loga-rithmic function has a base of e.

    Example 2

    Example 3

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  • Solving an Exponential Equation

    Solve and approximate the result to three decimal places.

    SolutionWrite original equation.

    Add 4 to each side.

    Divide each side by 2.

    Take log (base 3) of each side.

    Inverse Property

    Add 5 to each side.

    Divide each side by 2.

    Use a calculator.

    The solution is Check this in the original equation.

    Now try Exercise 53.

    When an equation involves two or more exponential expressions, you canstill use a procedure similar to that demonstrated in Examples 2, 3, and 4.However, the algebra is a bit more complicated.

    t 52 12 log3 7.5 3.417.

    t 3.417

    t 52

    12

    log3 7.5

    2t 5 log3 7.5

    2t 5 log3 152

    log3 32t5 log3

    152

    32t5 152

    232t5 15

    232t5 4 11

    232t5 4 11

    248 Chapter 3 Exponential and Logarithmic Functions

    Example 4Additional ExampleSolve

    Solution

    Note: Using the change-of-baseformula or the definition of a logarith-mic function, you could write thissolution as x log2 10.

    x ln 10ln 2

    3.322

    x ln 2 ln 10

    ln 2x ln 10

    2x 10

    2x 10.

    To ensure that students first solve forthe unknown variable algebraically andthen use their calculators, you canrequire both exact algebraic solutionsand approximate numerical answers.

    Remember that to evaluate alogarithm such as youneed to use the change-of-baseformula.

    log3 7.5 ln 7.5ln 3

    1.834

    log3 7.5,

    Solving an Exponential Equation of Quadratic Type

    Solve e2x 3ex 2 0.

    Example 5

    Algebraic SolutionWrite original equation.

    Write in quadratic form.

    Factor.

    Set 1st factor equal to 0.

    Solution

    Set 2nd factor equal to 0.

    Solution

    The solutions are and Checkthese in the original equation.

    Now try Exercise 67.

    x 0.x ln 2 0.693

    x 0

    ex 1 0

    x ln 2

    ex 2 0

    ex 2ex 1 0

    ex2 3ex 2 0

    e2x 3ex 2 0

    Graphical SolutionUse a graphing utility to graph Use thezero or root feature or the zoom and trace features of thegraphing utility to approximate the values of for which

    In Figure 3.25, you can see that the zeros occur atand at So, the solutions are and

    FIGURE 3.25

    3

    1

    3

    3y = e2x 3ex + 2

    x 0.693.x 0x 0.693.x 0

    y 0.x

    y e2x 3ex 2.

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  • Solving Logarithmic EquationsTo solve a logarithmic equation, you can write it in exponential form.

    Logarithmic form

    Exponentiate each side.

    Exponential form

    This procedure is called exponentiating each side of an equation.

    Solving Logarithmic Equations

    a. Original equation

    Exponentiate each side.

    Inverse Property

    b. Original equation

    One-to-One Property

    Add and 1 to each side.

    Divide each side by 4.

    c. Original equation

    Quotient Property of Logarithms

    One-to-One Property

    Cross multiply.

    Isolate

    Divide each side by

    Now try Exercise 77.

    Solving a Logarithmic Equation

    Solve and approximate the result to three decimal places.

    SolutionWrite original equation.

    Subtract 5 from each side.

    Divide each side by 2.

    Exponentiate each side.

    Inverse Property

    Use a calculator.

    Now try Exercise 85.

    x 0.607

    x e12

    eln x e12

    ln x 1

    2

    2 ln x 1

    5 2 ln x 4

    5 2 ln x 4

    7. x 2

    x. 7x 14

    3x 14 10x

    3x 14

    5 2x

    log63x 145 log6 2x log63x 14 log6 5 log6 2x

    x 2

    x 4x 8

    5x 1 x 7

    log35x 1 log3x 7

    x e2 e ln x e2

    ln x 2

    x e3

    eln x e3

    ln x 3

    Section 3.4 Exponential and Logarithmic Equations 249

    Example 6

    Example 7

    Remember to check your solu-tions in the original equationwhen solving equations to verifythat the answer is correct and tomake sure that the answer lies in the domain of the originalequation.

    Activities

    1. Solve for

    Answer:

    2. Solve for

    Answer: is not in thedomain.

    x 1 x 6logx 4 logx 1 1.

    x:

    x ln 3ln 7

    0.5646

    7x 3.x:

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  • Solving a Logarithmic Equation

    Solve 2

    SolutionWrite original equation.

    Divide each side by 2.

    Exponentiate each side (base 5).

    Inverse Property

    Divide each side by 3.

    The solution is Check this in the original equation.

    Now try Exercise 87.

    Because the domain of a logarithmic function generally does not include allreal numbers, you should be sure to check for extraneous solutions of logarithmicequations.

    x 253 .

    x 253

    3x 25

    5 log5 3x 52

    log5 3x 2

    2 log5 3x 4

    log5 3x 4.

    250 Chapter 3 Exponential and Logarithmic Functions

    Example 8

    Notice in Example 9 that thelogarithmic part of the equationis condensed into a single loga-rithm before exponentiatingeach side of the equation.

    In Example 9, the domain of is and the domain of isso the domain of the original equation is Because the domain is all

    real numbers greater than 1, the solution is extraneous. The graph inFigure 3.26 verifies this concept.

    x 4x > 1.x > 1,

    logx 1x > 0log 5x

    Checking for Extraneous Solutions

    Solve log 5x logx 1 2.

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