P Conol - 4.4 Study Guide, Checkup, & Quiz
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Waiakea High School *
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Course
MCA1040
Subject
Mathematics
Date
Nov 24, 2024
Type
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Uploaded by DukeDiscoveryCoyote7
Peyton Conol | 11.3.23 - 4.4 Study Guide, Checkup, & Quiz
Page 1:
a.
A logarithm is just a(n)
exponent
.
b.
y
=
lo g
b
x
is the same as
b
y
=
x
.
c.
lo g
8
64
=
?
is the same question as, "To what
exponent
do we need to raise
8
to get
64
."
Page 2:
Solve for the variable in each of the following equations.
a.
y
=
lo g
9
81
|
2
=
lo g
9
81
b.
3
=
lo g
5
x
|
3
=
lo g
5
15
c.
5
=
lo g
b
32
|
5
=
lo g
2
32
d.
y
=
lo g
3
243
|
5
=
lo g
3
243
e.
−
2
=
lo g
4
(
x
)
|
−
2
=
lo g
4
(
1
16
)
f.
−
3
=
lo g
b
(
1
216
)
|
−
3
=
lo g
6
(
1
216
)
Page 3:
a.
A common logarithm is any logarithm with a base of
10
, while a natural logarithm is any
logarithm with a base of
e
.
b.
If the base of a logarithm is not written, it is assumed to be
10
.
c.
The natural logarithm of x is written as
ln
=
x
.
d.
You should use the
log
button on a calculator to find the common logarithm of any
number greater than 0 and the
ln
button to find the natural log.
e.
What is the change in the base formula?
lo g
b
x
=
lo g
a
x
lo g
a
b
f.
The expression
lo g
4
256
can be written as a fraction with
log 256
as the numerator
and
log 4
as the denominator.
Page 4:
Calculate each of the following logarithms to 2 decimal places.
a.
lo g
7
102
=
¿
log102
log7
=
2.38
b.
lo g
3
44
=
¿
log 44
log 3
=
3.44
c.
lo g
6
13
=
¿
log13
log6
=
1.43
d.
lo g
2
275
=
¿
log275
log 2
=
8.1
Page 5:
Circle your answers where appropriate:
a.
lo g
b
MN
=
¿
lo g
b
M
+
lo g
b
N
b.
lo g
b
M
N
=
¿
lo g
b
M
−
lo g
b
N
c.
lo g
b
M
p
=
¿
plo g
b
M
d.
log
(
x
+
y
)
≠
logx
+
logy
e.
logx
logy
≠
log
x
y
Page 6:
Match the following expressions with an equivalent expression numbered i through vi.
a.
log
(
36
5
)
|
log 18
+
1
3
log 8
−
log 5
b.
ln
(
x
2
−
49
y
4
)
|
ln
(
x
−
7
)+
ln
(
x
+
7
)−
4ln
(
y
)
c.
x
+
49
¿
2
(
¿
)
18
x
6
¿
ln
¿
|
ln
(
18
)+
6 ln
(
x
)−
2ln
(
x
+
49
)
d.
2
lo g
3
(
x
)+
lo g
3
(
y
)+
lo g
3
(
z
)−
lo g
3
(
x
+
z
)
|
lo g
3
x
2
yz
x
+
z
e.
6
lo g
3
(
x
)+
2
lo g
3
(
x
+
z
)
|
x
+
z
¿
2
lo g
3
x
6
¿
f.
log
(
x
)+
2log
(
y
)+
log
(
z
)−
log
(
x
+
7
)
|
log
(
x y
2
z
x
+
7
)
Answer the following questions using what you've learned from this lesson. Write your responses
in the space provided.
1. Write the equation
y
=
b
x
in logarithmic form.
lo g
b
y
=
x
2. Write the equation
y
=
e
x
in logarithmic form.
ln
(
y
)=
x
3. Write the equation
y
= log
x
in exponential form.
10
y
=
x
4. Write the equation
y
= ln
x
in exponential form.
e
y
=
x
5. Show two ways of finding
y
if
y
= log
4
64. Explain your thinking.
4
?
=
64
y
=
3
3
=
lo g
4
64
4
y
=
64
4
y
=
4
3
y
=
3
6. Find
y
if
y
= log
32
1. Explain your thinking.
32
y
=
1
32
y
=
32
0
y
=
0
7. Use the rules of logarithms to expand
3
x
+
¿
¿
¿
log
¿
.
3
x
+
¿
¿
x
−
5
¿
7
log
¿
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4
lo g
10
(
x
+
3
)+
7
lo g
10
(
x
−
5
)
8. Use the rules of logarithms to expand
ln
a
3
b
5
c
8
.
ln
(
a
3
b
5
)−
ln
(
c
8
)
d
ln
(
a
3
)−
ln
(
b
5
)−
8ln
(
c
)
3ln
(
a
)+
5 ln
(
b
)−
8 ln
(
c
)
9. Write the expression
3log
(
x
−
2
)+
log
(
x
+
2
)−
log
(
x
2
−
4
)
as a single logarithm.
2
lo g
10
(
x
−
2
)
10. Write the expression 5ln
M
+ 3ln
N
- 2ln
P
- ln
R
as a single logarithm.
ln
(
M
5
N
3
P
2
R
)
11. Solve for
x
if 5 = log
3
(
x
2
+ 18).
x
2
+
18
=
3
5
x
2
+
18
=
243
x
2
=
225
x
=
±
15
12. Solve for
x
if -2 = log(3
x
+ 5).
3
x
+
5
=
10
−
2
3
x
+
5
=
1
10
2
3
x
=
1
100
−
5
x
=
−
499
300
13. Use the change-of-base formula and a calculator to evaluate log
5
65 to two decimal places.
lo g
5
65
=
log65
log5
=
2.59
14. Use the change-of-base formula and a calculator to evaluate log
8
125 to two decimal places.
lo g
8
125
=
log125
log 8
=
2.32
15. How much more intense is an earthquake that measures 7.5 on the Richter scale than one that
measures 6.2? (Recall that the Richter scale defines magnitude of an earthquake with the
equation
m
=
log
L
S
, where
I
is the intensity of the earthquake being measured, and
S
is the
intensity of a standard earthquake.)
M
=
log
L
S
10
M
=
I
S
I
=
10
(
7.5
−
6.2
)
I
=
19.95
16. The psychological sensation of loudness,
B
, is measured in decibels and is defined with
reference to the intensity,
I
0
, of a barely audible sound by the following equation. In this
equation,
I
is the intensity of the sound in question.
B
=
10log
I
I
0
If the sound of a jet engine during takeoff is 140 decibels and the sound of a rock concert is 115
decibels, how does the intensity of the sound of the jet engine compare to the intensity of the
rock concert?
B
=
10log
140
115
The jet engine is significantly intense compared to the rock concert.
17. True or false:
ln
(
x
+
y
)−
ln
(
x
+
y
)=
0
. Explain your thinking.
True, the equation cancels each other.
18. True or false:
log
(
x
+
y
)−
log
(
y
)=
log
(
x
)
. Explain your thinking.
False,
log
(
x
+
y
)
≠
logx
+
logy
19. True or false:
logx
¿
2
=
2
logx
¿
. Explain your thinking.
True,
lo g
b
M
p
=
¿
plo g
b
M
20. True or false:
ln56
ln7
=
ln 8
. Explain your thinking.
False,
logx
logy
≠
log
x
y
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