Chapter 8

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8.3 Evaluating Logarithms GOAL Evaluate logarithmic expressions, and approximate the logarithm of a number to any base. YOU WILL NEED • graphing calculator LEARN ABOUT the Math Jackson knows that a rumour spreads very quickly. He tells three people a rumour. By the end of the next hour, each of these people has told three more people. Each person who hears the rumour tells three more people in the next hour. Jackson has written an algebraic model, N(t) 5 3 t11 , to represent the number of people who hear the rumour within a particular hour, where N(t) is the number of people told during hour t and t 5 1 corresponds to the hour during which the first three people heard the rumour and started telling others. ? In which hour will an additional 2187 people hear the rumour? EXAMPLE 1 Selecting a strategy to solve a problem Determine the hour in which an additional 2187 people will hear the rumour. Solution A: Using a guess-and-check strategy to solve an exponential equation N(t) 5 2187 Substitute 2187 for N(t) in the 2187 5 3 t11 equation. It is easier to solve the equation if both sides are written as powers with the same base. Using guess and check, write 2187 as a power of 3. 3 7 5 3 t11 Both sides will be equal when both 7 5 t 1 1 powers of 3 have the same exponent. Equate the exponents, 6 5 t and solve for t. Another 2187 students will hear the rumour during the 6th hour. NEL <strong>Chapter</strong> 8 459
Solution B: Using a graphing calculator to solve an exponential equation N(t) 5 2187 2187 5 3 t11 A graph can be used to solve the equation. Enter y 5 3 x11 in Y1 of the equation editor and y 5 2817 in Y2. Graph using a window that corresponds to the domain and range in this situation. The point of intersection for the two functions is the solution to the equation. Use the intersect operation to determine this point. Another 2187 students will hear the rumour during the 6th hour. Solution C: Rewriting an exponential equation in logarithmic form N(t) 5 2187 2187 5 3 t11 t 1 1 5 log 3 2187 Determine the value of the exponent t, when N(t) 5 2187. To solve for t, rewrite the equation in logarithmic form. Since a logarithm is an exponent, evaluate log 3 2187 by determining the exponent to which the base 3 must be raised to get 2187. Use guess and check. t 1 1 5 7 t 5 6 Another 2187 students will hear the rumour during the 6th hour. 460 8.3 Evaluating Logarithms 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: PRACTISING 4. Let f (x) 5 log 10 x.
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 and 44: 8. Solve for x. a) 4 x11 1 4 x 5 16
Page 45 and 46: Reflecting A. What strategies for s
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
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d) D 5 5xPR 0 x . 06, R 5 5 yPR6 e)
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17. log b x!x 5 log b x 1 log b !x
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value of x. Determine the average r
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Chapter 8

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