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Section 4.7 Laws of Logarithms

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. x y. Exponent Law Product Law x a · x b = x a + b Quotient Law x a ÷ x b = x a - b Power Law (x a ) b = x ab.

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Section 4.7 Laws of Logarithms

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  1. Section 4.7 Laws of Logarithms

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

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

  4. EXAMPLE 1Change log to a form involving the operations of addition and subtraction. a2b c4 a2b c4 log log (a2b) – log c4 log a2 + log b – log c4 2 log a + log b – 4 log c

  5. EXAMPLE 2Calculate using logarithms. (3.49)12 (82)(4.27) (3.49)12 (82)(4.27) (3.49)12 (82)(4.27) x = log x = log log x = log (3.49)12 – log [(82)(4.27)] log x = log (3.49)12 – [log 82 + log 4.27]

  6. EXAMPLE 2Calculate using logarithms. (3.49)12 (82)(4.27) 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

  7. Practice:Calculate using logarithms. Round your answer to the nearest ten. 4.7(8.35)7 13.173 Answer: 5820

  8. EXAMPLE 3Find 57. 1 2 x = 57 1 2 1 2 log x = log 57 log x = log 57 log x ≈ 0.8779 x = 100.8779 x = 7.55

  9. 3 Practice:Find 81 using logarithms. Round your answer to the nearest thousandth. Answer 4.327

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

  11. log2 5.89 = log 5.89 log 2 EXAMPLE 4Find log2 5.89 ≈ 2.558

  12. Practice:Find log3 19.53. Round your answer to the nearest hundredth. Answer 2.71

  13. Homework pp. 206-207

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

  15. ►A. Exercises Change each expression to a form involving addition and subtraction of terms by applying the laws of logarithms. 3. log a4 b2

  16. ►A. Exercises Change each expression to a form involving addition and subtraction of terms by applying the laws of logarithms. 5. log x3y2z5

  17. ►A. Exercises Find the log of each number in the given base. 7. log3 3.78

  18. ►A. Exercises Evaluate the following problems using logarithms. Show your work. 11. (4.97)2(5.6)

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

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

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

  22. ►B. Exercises If loga 5 = P and loga 2 = Q, find the following. 21. loga 2a7 loga 2a7 = loga 2 + loga a7 = loga 2 + 7loga a = Q + 7

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

  24. ■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.

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

  26. ■Cumulative Review Without graphing, determine whether the following functions are even, odd, or neither. 27. g(x) = x2 + 4x +4

  27. ■Cumulative Review Without graphing, determine whether the following functions are even, odd, or neither. 28. h(x) = |x| + x2

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