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.

Product Law xa · xb = xa + b Quotient Law xa ÷ xb = xa - b 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 x y

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

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

EXAMPLE 2 Calculate 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

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

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

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

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

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

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

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 logarithms. 3. log a4 b2

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

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

►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. 15. 93 7

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

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

►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

■ Cumulative Review 24. Solve a tan 3x + 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) = x2 + 4x +4

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