Download 7-6 Properties of Logarithms

7-6
Properties of Logarithms
TEKS FOCUS
VOCABULARY
Foundational to TEKS (5)(D) Solve exponential equations of the form
y = abx where a is a nonzero real number and b is greater than zero and
not equal to one and single logarithmic equations having real solutions.
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including mental
math, estimation, and number sense as appropriate, to solve problems.
ĚChange of Base Formula – logb M =
where M, b, and c are positive numbers,
b ≠ 1, and c ≠ 0.
ĚNumber sense – the understanding of
what numbers mean and how they are
related
Additional TEKS (1)(A)
ESSENTIAL UNDERSTANDING
Logarithms and exponents have corresponding properties.
Properties
Properties of Logarithms
For any positive numbers m, n, and b where b ≠ 1, the following properties apply.
Product Property
log b mn = log b m + log b n
Quotient Property
log b n = log b m - log b n
Power Property
log b mn = n log b m
m
Here’s Why It Works You can use a product property of exponents to derive a
product property of logarithms.
Let x = log b m and y = log b n.
m = bx and n = by
mn =
298
Lesson 7-6
bx
#
by
Definition of logarithm
Write mn as a product of powers.
mn = bx+y
Product Property of Exponents
log b mn = x + y
Definition of logarithm
log b mn = log b m + log b n
Substitute for x and y.
Properties of Logarithms
logc M
,
logc b
Change of Base Formula
Property
You have seen logarithms with many bases. The log key on a calculator finds log 10
of a number. To evaluate a logarithm with any base, use the Change of Base Formula.
For any positive numbers m, b, and c, with b ≠ 1 and c ≠ 1,
log m
log b m = logc b .
c
Here’s Why It Works
log b m =
=
(log b m)(log c b)
log c b
Multiply logb m by
log c blogb m
log c b
Power Property of Logarithms
log c m
logc b
= 1.
logc b
blogb m = m
= log b
c
Problem 1
P
Simplifying Logarithms
What is each expression written as a single logarithm?
A log 4 32 − log 4 2
log 4 32 - log 4 2 = log 4 32
2
What must you do
with the numbers
that multiply the
logarithms?
Apply the Power Property
of Logarithms.
Quotient Property of Logarithms
= log 4 16
Divide.
= log 4 42
Write 16 as a power of 4.
=2
Simplify.
B 6 log 2 x + 5 log 2 y
6 log 2 x + 5 log 2 y = log 2 x6 + log 2 y 5
= log 2
x 6y 5
Power Property of Logarithms
Product Property of Logarithms
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Problem 2
P
Expanding Logarithms
What is each logarithm expanded?
4x
A log y
log 4x
y = log 4x - log y
Quotient Property of Logarithms
= log 4 + log x - log y
Product Property of Logarithms
x4
Can you apply the
Power Property of
Logarithms first?
No; the fourth power
applies only to x.
B log 9 729
x4
log 9 729
= log 9 x4 - log 9 729
Quotient Property of Logarithms
= 4 log 9 x - log 9 729
Power Property of Logarithms
= 4 log 9 x - log 9 93
Write 729 as a power of 9.
= 4 log 9 x - 3
Simplify.
Problem
bl
3
TEKS Process Standard (1)(C)
Using the Change of Base Formula
What is the value of each expression?
What common base
has powers that
equal 27 and 81?
3; 33 = 27 and
34 = 81.
A log 81 27
Method 1 Use a common base.
log 27
log 81 27 = log 3 81
3
= 34
Change of Base Formula
Simplify.
Method 2 Use a calculator.
log 27
log 81 27 = log 81
= 0.75
What would be a
reasonable result?
52 = 25 and 53 = 125,
so log 5 36 should be
between 2 and 3.
300
Lesson 7-6
Change of Base Formula
log(27)/log(81)
.75
Use a calculator.
B log 5 36
log 36
log 5 36 = log 5
≈ 2.23
Properties of Logarithms
Change of Base Formula
Use a calculator to evaluate.
log(36)/log(5)
2.226565505
Problem 4
P
TEKS Process Standard (1)(A)
Using a Logarithmic Scale
STEM
Chemistry The pH of a substance equals −log [H + ], where [H + ] is the
concentration of hydrogen ions. [H +a] for household ammonia is 10−11. [H +v]
for vinegar is 6.3 × 10−3. What is the difference of the pH levels of ammonia
and vinegar?
Write the equation
for pH.
pH = −log [H + ]
p
Write the difference of
the pH levels.
−log [H +a] − (−log [H +v])
Substitute values for
[H+v ] and [H+a].
= log (6.3 × 10−3) − log 10−11
Use the Product Property
of Logarithms, and simplify.
= log 6.3 + log 10−3 − log 10−11
= −log [H +a] + log [H +v]
= log [H +v] − log [H +a]
= log 6.3 − 3 + 11
Use a calculator.
? 8.8
The pH level of ammonia is about
8.8 greater than the pH level
of vinegar.
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Write the answer.
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Write each expression as a single logarithm.
1. log 7 + log 2
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completing your homework,
go to PearsonTEXAS.com.
4. log 8 - 2 log 6 + log 3
2. log 2 9 - log 2 3
5. 4 log m - log n
3. 5 log 3 + log 4
6. log 5 - k log 2
7. Apply Mathematics (1)(A) The loudness in decibels (dB) of a sound is defined
as 10 log II , where I is the intensity of the sound in watts per square meter
0
1 W>m2 2 . I0, the intensity of a barely audible sound, is equal to 10-12 W>m2. Town
regulations require the loudness of construction work not to exceed 100 dB. Suppose
a construction team is blasting rock for a roadway. One explosion has an intensity
of 1.65 * 10-2 W>m2. Is this explosion in violation of town regulations?
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Expand each logarithm.
8. log x3y 5
9. log 7 49xyz
13. log 3 (2x)2
12. log 5 rs
10. log b bx
14. log 3 7(2x - 3)2
11. log a2
15. log 5 25
x
Use the Change of Base Formula to evaluate each expression.
16. log 2 9
20. log 4 7
STEM
STEM
17. log 12 20
21. log 3 54
18. log 7 30
22. log 5 62
19. log 5 10
23. log 3 33
24. Apply Mathematics (1)(A) The concentration of hydrogen ions in
household dish detergent is 10-12. What is the pH level of household
dish detergent?
electron
–1
25. Apply Mathematics (1)(A) The foreman of a construction team puts
up a sound barrier that reduces the intensity of the noise by 50%. By
how many decibels is the noise reduced? Use the formula L = 10 log II
0
to measure loudness. (Hint: Find the difference between the expression
for loudness for intensity I and the expression for loudness for
intensity 0.5I.)
26. Create Representations to Communicate Mathematical Ideas
(1)(E) Explain why the expansion at the right of log 4 5 st is
incorrect. Then do the expansion correctly.
27. Explain Mathematical Ideas (1)(G) Can you expand
log 3 (2x + 1)? Explain.
28. Explain Mathematical Ideas (1)(G) Explain why log (5
log4
+1
proton
t
s
t
= 21 log4 s
= 21 log4t – log4S
# 2) ≠ log 5 # log 2.
Use the properties of logarithms to evaluate each expression.
29. log 2 4 - log 2 16
32. log 6 12 + log 6 3
30. log 2 96 - log 2 3
33. log 4 48 - 12 log 4 9
31. log 3 27 - 2 log 3 3
34. 12 log 5 15 - log 5 175
Determine if each statment is true or false. Justify your answer.
36. log 3 32 = 12 log 3 3
35. log 2 4 + log 2 8 = 5
log x
log x
37. log (x - 2) = log 2
38. logb y = log b xy
b
39. (log x)2 = log x2
40. log 4 7 - log 4 3 = log 4 4
Write each logarithmic expression as a single logarithm.
41. 14 log 3 2 + 14 log 3 x
42. 12 (log x 4 + log x y) - 3 log x z
43. x log 4 m + 1y log 4 n - log 4 p
44.
(
2 log b x 3 log b y
+
- 5 log b z
4
3
)
Write each logarithm as the quotient of two common logarithms. Do not simplify
the quotient.
45. log 7 2
302
Lesson 7-6
46. log 3 8
Properties of Logarithms
47. log 5 140
48. log 9 3.3
49. log 4 3x
STEM
Apply Mathematics (1)(A) The apparent brightness of stars is
measured on a logarithmic scale called magnitude, in which
lower numbers mean brighter stars. The relationship between
the ratio of apparent brightness of two objects and the
difference in their magnitudes is given by the formula
Capella
m = 0.1
b
m2 − m1 = −2.5 log b2 , where m is the magnitude and b is
1
the apparent brightness.
50. How many times brighter is a magnitude 1.0 star than a
magnitude 2.0 star?
51. The star Rigel has a magnitude of 0.12. How many times
brighter is Capella than Rigel?
Simplify each expression.
52. log 3 (x + 1) - log 3 (3x2 - 3x - 6) + log 3 (x - 2)
1
53. log (a2 - 10a + 25) + 12 log
3 - log (1a - 5)
(a - 5)
TEXAS Test Practice
T
54. Assume that there are no more turning points beyond those shown. Which graph
CANNOT be the graph of a fourth degree polynomial?
A.
C.
y
y
x
B.
x
D.
y
y
x
x
55. A florist is arranging a bouquet of daisies and tulips. He wants twice as many
daisies as tulips in the bouquet. If the bouquet contains 24 flowers, how many
daisies are in the bouquet?
F. 8 daisies
G. 12 daisies
H. 16 daisies
J. 24 daisies
56. Use the properties of logarithms to write log 18 in four different ways. Name each
property you use.
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