Section 4 7 Laws of Logarithms Objectives 1

Section 4. 7 Laws of Logarithms

Objectives: 1. To state and apply the laws of logarithms. 2. To use the change of base formula to find logarithms in any base.

Exponent Law Product Law Quotient Law xa · xb = xa + b xa ÷ xb = xa - b Power Law (xa)b = xab Laws of Logarithms Product Law logb xy = logb x + logb x y Quotient Lawlogb y y = logb x – logb

a 2 b EXAMPLE 1 Change log 4 to c a form involving the operations of addition a 2 b and subtraction. log 4 c log (a 2 b) – log c 4 log a 2 + log b – log c 4 2 log a + log b – 4

(3. 49)12 EXAMPLE 2 Calculate (82)(4. 27) using logarithms. (3. 49)12 x= (82)(4. 27) (3. 49)12 log x = (82)(4. 27) log x = log (3. 49)12 – log [(82)(4. 27)] log x = log (3. 49)12 – [log 82 +

(3. 49)12 EXAMPLE 2 Calculate (82)(4. 27) using logarithms. log x = log (3. 49)12 – [log 82 + log 4. 27] log x = 12 log (3. 49) – log 82 – log 4. 27 log x ≈ 3. 96966 x ≈ 103. 96966 x ≈ 9325

4. 7(8. 35)7 Practice: Calculate 13. 173 using logarithms. Round your answer to the nearest ten. Answer: 5820

EXAMPLE 3 Find 57. 1 2 x = 57 1 2 log x = log 1 57 log x =2 log 57 log x ≈ 0. 8779 x = 100. 8779 x = 7. 55

Practice: Find 81 using 3 logarithms. Round your answer to the nearest thousandth. Answer 4. 327

Change of base formula: logb x = loga x loga b

EXAMPLE 4 Find log 2 5. 89 = log 2 ≈ 2. 558

Practice: Find log 3 19. 53. Round your answer to the nearest hundredth. Answer 2. 71

Homework pp. 206 -207

►A. Exercises Change each expression to a form involving addition and subtraction of terms by applying the laws of logarithms. 1. log xy

►A. Exercises Change each expression to a form involving addition and subtraction of terms by applying the laws of 4 a logarithms. b 2 3. log

►A. Exercises Change each expression to a form involving addition and subtraction of terms by applying the laws of logarithms. 5. log x 3 y 2 z 5

►A. Exercises Find the log of each number in the given base. 7. log 3 3. 78

►A. Exercises Evaluate the following problems using logarithms. Show your work. 11. (4. 97)2(5. 6)

►A. Exercises Evaluate the following problems using logarithms. Show your work. 7 15. 93

►B. Exercises If loga 5 = P and loga 2 = Q, find the following. 17. loga 10= loga (2 ∙ 5) = loga 2 + loga 5 =Q+P

►B. Exercises If loga 5 = P and loga 2 = Q, find the following. 19. loga 2 = 1 2 loga 2 1 =2 Q

►B. Exercises If loga 5 = P and loga 2 = Q, find the following. 21. loga 2 a 7 = loga 2 + loga a 7 = loga 2 + 7 loga a =Q+7

■ Cumulative Review 24. Solve a tan 3 x + b = c for x

■ Cumulative Review 25. Write the equations of the natural log function and its inverse, where each of them has been translated left 2 units and down 3 units.

■ Cumulative Review Without graphing, determine whether the following functions are even, odd, or neither. 26. f(x) = sin x

■ Cumulative Review Without graphing, determine whether the following functions are even, odd, or neither. 27. g(x) = x 2 + 4 x +4

■ Cumulative Review Without graphing, determine whether the following functions are even, odd, or neither. 28. h(x) = |x| + x 2