Chapter 8

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8.6 Solving Logarithmic Equations GOAL Solve logarithmic equations with one variable algebraically. LEARN ABOUT the Math The Richter scale is used to compare the intensities of earthquakes. The Richter scale magnitude, R, of an earthquake is determined using R 5 log Q a , where a is the amplitude of the vertical ground T R 1 B motion in microns ( m), T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of the seismic waves. ( 1 m is equivalent to 10 26 m. ) ? An earthquake measured 5.5 on the Richter scale, and the period of the seismic wave was 1.8 s. If B equals 3.2, what was the amplitude, a, of the vertical ground motion? EXAMPLE 1 Selecting an algebraic strategy to solve a logarithmic equation Determine the amplitude, a, of the vertical ground motion. Solution R 5 log a a T b 1 B 10 2.3 5 a 1.8 10 2.3 3 1.8 5 a 5.5 5 log a a 1.8 b 1 3.2 2.3 5 log a a 1.8 b 359.1 m 5a The amplitude of the vertical ground motion was about 359.1 m. Substitute the given values into the equation. Isolate the term with the unknown, a, by subtracting 3.2 from both sides. Rewrite the equation in exponential form. Multiply both sides by 1.8 to solve for a. To get a better idea of the size of this number, change microns to metres or centimetres. 359.1 m 50.000 359 1 m or 0.035 91 cm. NEL <strong>Chapter</strong> 8 487
Reflecting A. What strategies for solving a linear equation were used to solve this logarithmic equation? B. Why was the equation rewritten in exponential form? C. How would the strategies have changed if the value of a had been given and the value of T had to be determined? APPLY the Math EXAMPLE 2 Selecting an algebraic strategy to solve a logarithmic equation Solve. a) log x 0.04 522 b) Solution log 7 (3x 2 5) 5 log 7 16 a) log x 0.04 522 x 22 5 0.04 Express the equation in exponential form. x 22 5 1 25 x 22 5 5 22 x 5 5 Rewrite the decimal as a fraction. 0.04 5 4 100 5 1 25 1 Express as a power with 25 exponent 22. Since the exponents are equal, the bases must be equal. b) log 7 (3x 2 5) 5 log 7 16 3x 2 5 5 16 3x 5 21 x 5 7 If log a M 5 log a N, then M 5 N. Since 7 is the base of both logs, the two expressions must be equal. Add 5 to both sides of the equation. 488 8.6 Solving Logarithmic Equations NEL
Page 1 and 2: 444 NEL
Page 3 and 4: 8 Getting Started Study Aid • For
Page 5 and 6: 8.1 Exploring the Logarithmic Funct
Page 7 and 8: In Summary Key Ideas • The invers
Page 9 and 10: 8.2 Transformations of Logarithmic
Page 11 and 12: Reflecting L. Describe the domain a
Page 13 and 14: EXAMPLE 2 Connecting a geometric de
Page 15 and 16: PRACTISING 4. Let f (x) 5 log 10 x.
Page 17 and 18: Solution B: Using a graphing calcul
Page 19 and 20: c) log 2 (24) 5 x Determine the exp
Page 21 and 22: EXAMPLE 4 Selecting a strategy to e
Page 23 and 24: CHECK Your Understanding 1. Express
Page 25 and 26: 17. The doubling function y 5 y 0 2
Page 27 and 28: log a (mn) 5 x 1 y m 5 a x so log a
Page 29 and 30: APPLY the Math EXAMPLE 4 Selecting
Page 31 and 32: EXAMPLE 6 Selecting strategies to s
Page 33 and 34: 9. Write each expression as a singl
Page 35 and 36: Study • See Lesson 8.2, Examples
Page 37 and 38: 8.5 Solving Exponential Equations G
Page 39 and 40: Solution C: Solving the equation gr
Page 41 and 42: EXAMPLE 4 Solve 2 x11 5 3 x21 to th
Page 43: 8. Solve for x. a) 4 x11 1 4 x 5 16
Page 47 and 48: x 2 2 9 5 2 4 x 2 2 9 5 16 x 2 5 25
Page 49 and 50: 9. The loudness, L, of a sound in d
Page 51 and 52: APPLY the Math EXAMPLE 1 Solving a
Page 53 and 54: EXAMPLE 2 Representing exponential
Page 55 and 56: Solution Sound Loudness (dB) soft w
Page 57 and 58: 6. Calculate to two decimal places
Page 59 and 60: 8.8 Rates of Change in Exponential
Page 61 and 62: EXAMPLE 2 Selecting a strategy for
Page 63 and 64: Solution -2 Tangents to y = 10 x y
Page 65 and 66: y 8 y = log x - 1 6 2 d 4 2 y = - 1
Page 67 and 68: PRACTICE Questions Lesson 8.1 1. De
Page 69 and 70: 8 Chapter Self-Test 1. Write the eq
Page 71 and 72: 2(sec 2 x 2 tan 2 x) 9. 5 sin 2x se
Page 73 and 74: d) D 5 5xPR 0 x . 06, R 5 5 yPR6 e)
Page 75 and 76: 17. log b x!x 5 log b x 1 log b !x
Page 77: value of x. Determine the average r

Chapter 8

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