Section 4.7 Laws of Logarithms
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Section 4.7 Laws of Logarithms
description
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 anybase. x y. Exponent Law Product Lawx a · x b = x a + b Quotient Lawx a ÷ x b = x a - b Power Law(x a ) b = x ab. - PowerPoint PPT Presentation
Transcript of Section 4.7 Laws of Logarithms
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 anybase.
Objectives:1. To state and apply the laws of
logarithms.2. To use the change of base
formula to find logarithms in anybase.
Exponent Law
Product Law xa · xb = xa + b
Quotient Law xa ÷ xb = xa - b
Power Law (xa)b = xab
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
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
xyxy
EXAMPLE 1 Change log to a
form involving the operations of addition and subtraction.
EXAMPLE 1 Change log to a
form involving the operations of addition and subtraction.
a2bc4
a2bc4
logloga2bc4
a2bc4
log (a2b) – log c4
log a2 + log b – log c4
2 log a + log b – 4 log c
log (a2b) – log c4
log a2 + log b – log c4
2 log a + log b – 4 log c
EXAMPLE 2 Calculate using logarithms. EXAMPLE 2 Calculate using logarithms.
(3.49)12
(82)(4.27)(3.49)12
(82)(4.27)
x =x =(3.49)12
(82)(4.27)(3.49)12
(82)(4.27)
log x = loglog x = log(3.49)12
(82)(4.27)(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]
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. EXAMPLE 2 Calculate using logarithms.
(3.49)12
(82)(4.27)(3.49)12
(82)(4.27)
log x = log (3.49)12 – [log 82 + log 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
log x = 12 log (3.49) – log 82 – log 4.27
log x ≈ 3.96966
x ≈ 103.96966
x ≈ 9325
Answer: 5820Answer: 5820
Practice: Calculate using
logarithms. Round your answer to the nearest ten.
Practice: Calculate using
logarithms. Round your answer to the nearest ten.
4.7(8.35)7
13.173
4.7(8.35)7
13.173
EXAMPLE 3 Find 57.EXAMPLE 3 Find 57.
x = 57x = 571212
log x = log 57log x = log 571212
log x = log 57log x = log 571212
log x ≈ 0.8779
x = 100.8779
x = 7.55
log x ≈ 0.8779
x = 100.8779
x = 7.55
Answer 4.327Answer 4.327
Practice: Find 81 using logarithms. Round your answer to the nearest thousandth.
Practice: Find 81 using logarithms. Round your answer to the nearest thousandth.
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Change of base formula:Change of base formula:
logb x =logb x =loga x
loga b
loga x
loga b
EXAMPLE 4 Find log2 5.89EXAMPLE 4 Find log2 5.89
log2 5.89 =log2 5.89 =log 5.89
log 2
log 5.89
log 2
≈ 2.558≈ 2.558
Answer 2.71Answer 2.71
Practice: Find log3 19.53. Round your answer to the nearest hundredth.Practice: Find log3 19.53. Round your answer to the nearest hundredth.
Homework
pp. 206-207
Homework
pp. 206-207
►A. ExercisesChange each expression to a form involving addition and subtraction of terms by applying the laws of logarithms.
1. log xy
►A. ExercisesChange each expression to a form involving addition and subtraction of terms by applying the laws of logarithms.
1. log xy
►A. ExercisesChange each expression to a form involving addition and subtraction of terms by applying the laws of logarithms.
3. log
►A. ExercisesChange each expression to a form involving addition and subtraction of terms by applying the laws of logarithms.
3. loga4
b2
a4
b2
►A. ExercisesChange each expression to a form involving addition and subtraction of terms by applying the laws of logarithms.
5. log x3y2z5
►A. ExercisesChange each expression to a form involving addition and subtraction of terms by applying the laws of logarithms.
5. log x3y2z5
►A. ExercisesFind the log of each number in the given base.
7. log3 3.78
►A. ExercisesFind the log of each number in the given base.
7. log3 3.78
►A. ExercisesEvaluate the following problems using logarithms. Show your work.11. (4.97)2(5.6)
►A. ExercisesEvaluate the following problems using logarithms. Show your work.11. (4.97)2(5.6)
►A. ExercisesEvaluate the following problems using logarithms. Show your work.15. 93
►A. ExercisesEvaluate the following problems using logarithms. Show your work.15. 9377
►B. ExercisesIf loga 5 = P and loga 2 = Q, find the following.17. loga 10
►B. ExercisesIf loga 5 = P and loga 2 = Q, find the following.17. loga 10
loga 10 = loga (2 ∙ 5)= loga 2 + loga 5= Q + P
loga 10 = loga (2 ∙ 5)= loga 2 + loga 5= Q + P
►B. ExercisesIf loga 5 = P and loga 2 = Q, find the following.19. loga 2
►B. ExercisesIf loga 5 = P and loga 2 = Q, find the following.19. loga 2
loga 2 = loga 2loga 2 = loga 21212
= loga 2= loga 21212
= Q= Q1212
►B. ExercisesIf loga 5 = P and loga 2 = Q, find the following.21. loga 2a7
►B. ExercisesIf 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
loga 2a7 = loga 2 + loga a7
= loga 2 + 7loga a= Q + 7
■ Cumulative Review24. Solve a tan 3x + b = c for x
■ Cumulative Review24. Solve a tan 3x + b = c for x
■ Cumulative Review25. 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 Review25. 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 ReviewWithout graphing, determine whether the following functions are even, odd, or neither. 26. f(x) = sin x
■ Cumulative ReviewWithout graphing, determine whether the following functions are even, odd, or neither. 26. f(x) = sin x
■ Cumulative ReviewWithout graphing, determine whether the following functions are even, odd, or neither. 27. g(x) = x2 + 4x +4
■ Cumulative ReviewWithout graphing, determine whether the following functions are even, odd, or neither. 27. g(x) = x2 + 4x +4
■ Cumulative ReviewWithout graphing, determine whether the following functions are even, odd, or neither. 28. h(x) = |x| + x2
■ Cumulative ReviewWithout graphing, determine whether the following functions are even, odd, or neither. 28. h(x) = |x| + x2
