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. 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

4. 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

5. 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]

6. 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 ≈
x ≈
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 3 Find 57. x = 57 log x = log 57 log x = log 57 log x ≈ 0.87791 2
log x = log 57 1 2
log x = log 57 1 2
log x ≈
x =
x = 7.55

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

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

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

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

13. Homework pp

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.

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. 7

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

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