Properties of Logarithms
7-4
Vocabulary
Review
1. Circle the logarithms below.
39
log4 64
log (x 2 2 1)
a ? b(x11)
2. Circle the base of each logarithm
log2 32
log7 49
log10 57
log5 125
F
3. If logb x 5 y, then b x 5 y.
T
4. If log x 5 w, then 10w 5 x.
F
5. If 32 5 x, then log2 x 5 3.
Vocabulary Builder
formula (noun)
FAWRM
yoo luh
Other Word Forms: formulate (verb), formulaic (adjective)
Definition: A mathematical formula is an equation that you can use to solve a
particular kind of problem.
Example: You can use the quadratic formula to solve quadratic equations. You can
use the distance formula to find the distance between two points.
Use Your Vocabulary
6. Circle the slope formula.
A 5 12 (b1 1 b2)h
Chapter 7
A 5 Pert
P 5 2l 1 2w
198
y2 2 y1
m5x 2x
2
1
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Write T for true or F for false.
Since y 5 logax if and only if x 5 a y , logarithms and exponents have corresponding
properties.
Properties Properties of Logarithms
For any positive numbers m, n, and b where b 2 1, the following properties apply.
Product Property
Quotient Property
Power Property
logb mn 5 logb m 1 logb n
m
logb n 5 logb m 2 logb n
logb mn 5 n logb m
Use the properties of logarithms to complete each equation.
7. log 20 5 log (5 ? 4) 5 log 5 1 log 4
8. log5 (x 9) 5
9 log5x
24
9. log2 24 2 log2 3 5 log2
5 log 2 8
3
10. log3(5x 4) 5 log3 5
1 log3 x 4 5 log3 5
1 4 log3x
Problem 1 Simplif ying Logarithms
Got It? What is 2 log 4 6 2 log 4 9 written as a single logarithm? If possible, simplify
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
the single logarithm.
11. Circle the property you can use to rewrite 2 log 4 6.
Product Property
Quotient Property
Power Property
12. Use the property you circled above to rewrite the first term of 2 log 4 6 2 log 4 9.
2 log 4 6 2 log 4 9 5 log 4 36 2 log 4 9
13. Circle the property you can use to combine the last two terms in Exercise 12.
Product Property
Quotient Property
Power Property
14. Use the property you circled above to combine the two terms.
36
log 4 36 2 log 4 9 5 log 4
5 log 4 4
9
15. Use the definition of logarithm to simplify the expression.
log 4 4
5
Problem 2
1
Expanding Logarithms
Got It? What is log3 250
37 expanded? Simplify your answer, if possible.
199
Lesson 7-4
16. Follow the steps to expand the logarithm.
log3 250
37 5 log3
250
37
2 log3
Use the Quotient Property of Logarithms.
5 log3 2 1 log3
125
2 log3
37
Use the Product Property of Logarithms.
5 log3 2 1 log3
53
2 log3
37
Write 125 as a power of 5.
3 log3 5 2 log3
5 log3 2 1
37
Use the Power Property of Logarithms.
Properties Change of Base Formula
log m
For any positive numbers m, b, and c, with b u 1 and c u 1, logb m 5 logc b .
c
Use the Change of Base Formula to complete each equation.
log 100
17. log5 100 5
18. log2 100 5
log 5
log5 100
19. log
log5 2
log 100
100 5 log2 3
3
2
20. Reasoning The base implied in Exercise 17 is 2 / 5 / 10 / 100 .
Using the Change of Base Formula
Problem 3
Got It? What is the value of log8 32?
21. Circle the least common factor of 8 and 32.
4
8
32
22. Complete each equation.
Since 25 5 32, log2 32 5
log2 32
23. log8 32 5
Since 23 5 8, log2 8 5 3 .
5 .
5
5
log
2
8
3
Got It? What is the value of log 4 18?
24. Circle the calculator-ready form of log 4 18.
log 4
log 18
log 9
log 2
25. The value of log 4 18 is approximately
Problem 4
log 9
log 4
2.085
log 18
log4
.
Using a Logarithmic Scale
Got It? Chemistry The pH of a substance equals 2log fH1g, where fH1g is
the concentration of hydrogen ions. Suppose the hydrogen ion concentration for
Substance A is twice that for Substance B. Which substance has a greater pH level?
What is the greater pH level minus the lesser pH level? Explain.
Chapter 7
200
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
2
26. If fH 1bg is the concentration of hydrogen ions for Substance B, circle the pH
of Substance B.
2 log fH 1bg
log 12 fH 1bg
log f2H 1bg
2log fH 1bg
27. Circle the expression for the concentration of hydrogen ions for Substance A.
fH 1bg 2 2
fH1bg
2
2 ? fH 1bg
log fH 1bg
28. Circle the expression for the pH of Substance A.
2log f2 ? fH 1bgg
221 log fH 1bg
2 log f2fH 1bgg
2log fH 1bg
29. Use the Product Property of Logarithms to expand the expression you circled above.
2log f2 ? fH1b gg
5 2Q log 2 1 log fH 1bg R
5 2 0.301 2 log fH 1bg
(pH of Substance A) 5 2 0.301 1 (pH of Substance B)
30. Circle the substance with the greater pH level.
Substance A
Substance B
31. What is the difference between the pH levels? Explain how you know.
log 2, or 0.301. Explanations may vary. Sample: The log of twice a
_______________________________________________________________________
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
quantity x is log 2 more than log x.
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
Reasoning If log x 5 5, what is the value of 1x ?
32. Underline the correct expression to complete each sentence.
If log x 5 5, then x 5 105 > 510 . Since 1x 5 x 0 > x 21 , I know 1x 5 1025 > 5210 .
Math Success
Check off the vocabulary words that you understand.
logarithm
Change of Base Formula
Rate how well you can use the properties of logarithms.
Need to
review
0
2
4
6
8
Now I
get it!
10
201
Lesson 7-4