Algebra 2 Unit 8 (Chapter 7)
CALCULATORS ARE NOT ALLOWED
1.
Graph exponential functions. (Sections 7.1, 7.2)
Worksheet 1
1 – 36
2.
Solve exponential growth and exponential decay problems. (Sections 7.1, 7.2)
Worksheet 2
1 – 18
3.
Simplify logarithmic expressions. (Section 7.4)
Page 503
8 – 19
Worksheet 3
1 – 37
4.
Common and natural logs, inverse properties of
log x x w and x
graph logarithmic equations.
Worksheet 4
1 – 39
5.
Apply the 3 laws (properties) of logs. (Section 7.5)
Page 510
15 – 44
Worksheet 5
1 – 30
6.
Approximating logarithmic values. Change of base theorem.
Worksheet 6
1 – 23
7.
Solve exponential equations with a common base.
Worksheet 7
1-20
Page 519
3 – 11
8.
Solve logarithmic equations (Section 7.6)
Worksheet 8
1 – 34
9.
Solve log equations. (Section 7.6)
Worksheet 9
10.
1 – 33
Solve exponential equations without a common base.
Worksheet 10
1-13
Review
Review Worksheet 1
Review Worksheet 2
Review Worksheet 3
1 – 90
1 – 53
1 - 13
-1-
log x w
,
log b 1 ,
Algebra 2 Unit 8 Worksheet 1
CALCULATORS ARE NOT ALLOWED
Simplify:
3
4
5.
7
0
9.
-2
1.
5
2
2.
1
6.
4
36
10.
−3
1
2
The function defined by y = b
b > 0,
7.
1
 
2
11.
Definition:
Requirements:
3.
1
 
5
b
x
64
2
4.
1
 
4
8.
9 −2
−4
2
3
16
12.
−
− 12
1
2
is called an exponential function with base b
≠ 1
Characteristics of exponential functions:
The basic graph of an exponential function looks like the following:
Increasing
An increasing exponential if they rise as they go from left to right.
Decreasing
A decreasing exponential if they drop as they go from left to right.
Other characteristics: The x-axis is a horizontal asymptote of the graph and the
graphs contain the point (0,1).
In problems 13 – 16, complete the table of values and then graph on graph paper.
y= 2
13.
x
y
x
14.
x
y
y=3
x
1
15. y =  
2
x
y
x
1
16. y =  
5
x
2
2
2
2
1
1
1
1
0
0
0
0
–1
–1
–1
–1
–2
–2
–2
–2
-2-
y
x
Sketch the following graphs of the exponential functions and state if they are
increasing or decreasing graphs. Be sure to label the intercepts.
17. y = 4
x
1
18. y =  
3
x
19. y = 5
1
20. y =  
4
x
x
Create a table of values in problems 21 -23 and then graph on graph paper.
21.
y=1
x
22.
y= 0
x
23.
y = (–2)
x
24. Explain why the graphs of #21-23 are not exponential functions. What in the
equations is wrong?
Answer the following multiple choice questions based on your knowledge of
exponential functions and their graphs. Pay attention to increasing and decreasing
equations.
25.
x
If the equation of y = 5 is graphed, which of the following values of x would
produce a point closest to the x-axis?
2
7
a. 0
b. –1
c.
d.
3
4
x
26.
1
If the equation of y =   is graphed, which of the following values of x would
2
produce a point closest to the x-axis?
3
5
8
1
a.
b.
c.
d.
4
3
3
4
x
27.
1
If the equation of y =   is graphed, which of the following values of x would
3
produce a point closest to the x-axis?
7
2
a. 0
b. –1
c.
d.
4
3
28.
Which multiple choice ordered pair represents the y-intercept for the function
y = 2x ?
a. (0,0)
29.
b. (0, 1)
c. (0, 2)
d. there is no y-intercept
Select the correct multiple choice response.
The graph of y = 5 x lies in which quadrants?
a. Quadrants 1 and 2
b. Quadrants 1 and 3
-3-
c. Quadrants 1 and 4
30.
Select the correct multiple choice response.
x
 1 
The graph of y =   contains which of these points?
 10 
a. (0, 0)
31.
b. (0, 10)
c. (0, 1)
d. (0,
1
)
10
Which multiple choice ordered pair represents the x-intercept for the function
y= 4x ?
a. (0, 0)
b. (0, 1)
c. (1, 0)
d. there is no x-intercept
32.
Use the graph of y = 2 to answer the following multiple choice question.
x
If the equation y = 2 is graphed, which of the following values of x would
produce a point closest to the x-axis?
3
5
8
1
a.
b.
c.
d.
4
3
3
4
33.
Given the expression x where x > 1 and
statement is true?
x
n
n
a. the value of x = 0
n > 1, which multiple choice
n
b. the value of x > 0
n
n
c. the value of x < 0
d. the value of x = 1
34.
Given the expression x
statement is true?
a. the value of x
n
n
n
where x > 1 and
n = 0, which multiple choice
n
c. the value of x < 0
where 0 < x < 1 and
n > 1, which multiple choice
b. the value of x > 0
=0
n
d. the value of x = 1
35.
36.
n
Given the equation y = x
statement is true?
a. y = 0
b. y > 0
Given the equation y = x
statement is true?
a. y = 0
b. y > 0
c. y < 0
n
where x > 1 and
c. y < 0
-4-
d. y = 1
n < 0, which multiple choice
d. y = 1
Algebra 2 Unit 8 Worksheet 2
CALCULATORS ARE NOT ALLOWED
Many real world phenomena can be modeled by functions that describe how things grow or
decay as time passes. Examples of such phenomena include the studies of populations,
bacteria, the AIDS virus, radioactive substances, electricity, temperatures and credit
payments, to mention a few.
Any quantity that grows or decays by a fixed percent at regular intervals is said to possess
exponential growth or exponential decay.
Such a situation is called
Such a situation is called
Exponential Decay.
Exponential Growth.
The time required for a substance to decay and fall to one half of its initial value is
called the half-life. Radio-isotopes of different elements have different half-lives.
Some people are frightened of certain medical tests because the tests involve the
injection of radioactive materials. Doctors use isotopes whose radiation is extremely
low-energy, so the danger of mutation is very low. The half-life is long enough that the
doctors have time to take pictures, but not so long as to pose health problems. They
use elements that are not readily absorbed by the body but are voided or flushed long
before they get a change to decay within your body.
-5-
t
For the following word problems we will be using the exponential equation y = A (b ) h .
Where
1.
A is the initial amount
b is the amount of growth (or decay) that occurs in h time
t is time
Technetium-99m is one of the most commonly used radioisotopes for medical
purposes. It has a half life of 6 hours.
If 0.5 cc’s (which is less than a teaspoon) of Technetium-99m is injected for
a scan of a gallbladder, how much radioactive material will remain after 24 hours?
t
Use the formula
 1 6
y= A  
2
where A = the number of cc’s present initially
t = time in hours
2.
When a plant or animal dies, it stops acquiring Carbon-14 from the atmosphere.
Carbon-14 decays over time with a half-life of 5730 years.
How much of a 10mg sample will remain after 11,460 years?
t
Use the formula
 1 h
N = N0  
2
where N0 is the initial amount
N = the amount remaining
t = time in years
h = half life
3.
One certain element has a half-life of 1600 years. If 300 grams were present
originally, how many grams will remain after 3200 years?
4.
5.
6.
The radioactive gas radon has a half-life of 3 days. How much of an 80 gram
sample will remain after 9 days?
1
The radioactive gas radon has a half-life of approximately 3 days. About how
2
much of a 200 gram sample will remain after 1 week?
The population of a certain country doubles in size every 60 years. The
population is now 1 million people. Find its size in 180 years.
y = A (2)
t
60
A = initial population
t = time elapsed in years
-6-
7.
8.
9.
10.
Bacteria populations tend to have exponential growth rather than decay.
Suppose a certain bacteria population doubles in size every 12 hours. If you start
with 100 bacteria how many will there be in 48 hours?
A certain population of bacteria doubles every 3 weeks. The number of bacteria
now is only 10. How many will there be in 15 weeks?
A culture of yeast doubles in size every 20 minutes. The size of the culture is
now 70. Find its size in 1 hour (remember to convert 1 hour to minutes.)
The growth of a town doubles every year. If there are 64,000 people after 4
years, find the initial population.
11.
12.
The number of people with a flu virus is growing exponentially with time as
shown in the table below.
Flu Virus Growth
Day Number of People
0
400
1
800
2
1600
Which multiple choice equation expresses the number of bacteria, N, present
at any time, x ?
N=
d.
N = 400 • 2
b.
x
N = 400 + 2
c.
N = 800 • 2
–x
x
In the early years of the century the national debt was growing exponentially
with time as shown in the table below.
National Debt
Year Debt
0
30,000
1
60,000
2
120,000
Which multiple choice formula expresses the debt, y, at any time t ?
a.
13.
400
x
a.
t
y = 30,000 •2
b.
t
y = 10000 • 3
t
t
c.
y = 3 • 10
d.
y = 30,000 + 2
An epidemic of bubonic plague grew exponentially by the formula
t
A = A0 • 2
where
A0 = original amount infected
t = time passed in weeks
If 512,000 people were infected after 8 weeks, find the original amount
that were infected.
-7-
Use estimation for the following multiple choice questions:
Choose the best multiple choice response for the following:
14.
3
a.
15.
1.3
(12)•(2)
a.
16.
9
4
b.
3.9
c.
11.8
d.
35.5
b.
33.9
c.
67.3
d.
117.5
3
2
16.9
A radioactive element decays over time according to the equation: y = A
1
 
2
t
300
If 1000 grams were present initially, how may grams will remain after 650 years?
a.
17.
444
b.
222
c.
111
Boogonium decays using the formula: A = I • 2
d.
−t
h
55.5
The half life of Boogonium is 4 hours. How much of a 24 gram sample will remain
after 6 hours.
Choose the best multiple choice response.
a.
0.4
b.
3.2
c.
8.5
d.
16.9
18.
Geekonium-25 decays using the formula: A = I • 2
−t
h
The half life of Geekonium-25 is 2 years. Find how much of a 160 gram
sample remains after 8 years.
Unit 8 Worksheet 3
Determine the exponent needed to change the left number into the right
number. You may use positive, negative, zero, and fractional exponents.
Guess and Check:
1. 5 → 25
2. 4 → 64
3. 2 → ½
4. 3 → 1/9
5. 6 → 1
6. 27 → 3
7. 5 → 1/125
8. 16 → 4
9. 8 → 2 2
-8-
Logarithms (or logs) are used to find the exponents to help us solve
exponential equations.
Structure of a logarithm: log y = x
b
log28 = ?
Example: Simplify
b is the base
y is the value
x is the exponent on b to
yield y
=3 (because 23 = 8)
Simplify #10-29.
10. log6 36
11. log2 16
12. log10 100
1
13. log3  
9
14. log2 2 2
15. log7 1
16. log5 125
17. log4 16
18. log3 81
19. log6 6
20. log3 1
21. log8 4
 1 
22. log 5 

 25 
1
23. log 2  
8
24. log 6 6 6
25. log 5 25 5
26. log 7
27. log 3
28. log 2  3
3
49
5
9
 1

 4
 1 

 100 
29. log10 
Logarithms with a base 10 are called common logarithms. The base of 10 is implied and
not shown.
For example, log 1000 is equivalent to log 10 1000
Simplify: (Remember, when no base is given it is assumed to be base 10)
 1 

 10 
30. log 100
31. log 
32. log 1
33. log 10
34. log 0.01
35. log 10 3
36. log 0.0001
37. log
-9-
100
Unit 8 Worksheet 4
Log Rules :
log b y = x
(b
and y must be positive numbers, b
blog b x
bx = y
≠1)
log b b y = y
= x
π ≈ 3.14
log b 1 = 0
e ≈ 2.718
Remember, if no base is shown assume it is base 10, the common log.
log y = x
10 x = y
log 10 y = x
If base e is used it is called a natural log. Instead of writing log we use ln
log e y = x
ln y = x
ex = y
(Remember, e is just an irrational number. It is approximately 2.718; see Page 492 in
your textbook)
Restrictions:
You can’t take
log 0
or
With bases, you can’t do
log (of a negative number)
log 0 base or log 1 base or log negative base
Verify the log by rewriting the equation into exponential form.
1
1. log2 32 = 5
2. log3 9 = 2
3. log7 7 =
2
4. log3
Rewrite the equation in logarithmic form.
3
5. 4 = 64
3
2
7. 10
6. 9 = 27
−2
= 0.01
8. 16
Simplify:
9.
log 5 23
5
log9
13. log 7 49
14. log 8 64
17. log 8 1
18. log 7 7 5
12. log12 1
11. 10
10. log 2 27
x
15. log 4 16
19.
3log3 11
- 10 -
x
16. log 2 16
20. log 6 6
x
−
1
=–4
81
3
4
=
1
8
Write each equation in exponential form.
21. ln 8 = 2.08
22. ln 100 = 4.61
24. log1000 = 3
23. logπ 9.86 = 2
25. ln 1097 = 7
To graph a log equation: 1. First rewrite it in exponential form
2. Make a table of values. Look at the equation and see which
letter (x or y) is the exponent and put the numbers
2, 1, 0, –1, –2 in that column.
3. Plot the points and connect with a curve
Graph #26-29 on graph paper. Be sure to show the table of values and the
exponential equation.
26. y = log 2 x
27. y = log 5 x
28. y =
log 1 x
4
29. Graph y =
3x
and
y=
log 3 x
on the same grid.
Choose the correct multiple choice.
30.
a.
1
2
16 = 4 ?
Which is equivalent to
1
log 4   = 16
2
b.
1
log 16  
2
31.
Which is equivalent to logm n = p
32.
= n
Which is equivalent to log k = w
a.
mn = p
10 w = k
x
Given: y = 5
b. m
a.
33.
34.
= 4
log 16 4 =
1
2
d. log 4 16 =
p
n
c. n = m
d. p = m
?
1w = k
c. k
w
= 10
d.
10 k = w
which statement is true?
a.
y > 0 for all values of x
b. y > 0 for all values of x
c.
y < 0 for all values of x
d. y < 0 for all values of x
When is the following statement true?
a. for all values of x
1
2
?
p
b.
c.
7 log 7 x
b. for some values of x
d. can’t determine
- 11 -
=
x
c. for no values of x
35.
In the equation
log x y = z
a. z must always be positive
which statement is true about the value of z ?
b. z can never equal 0
c. z can never equal 1
d. there are no restrictions on z
36.
When is the equation
log 6 6y = y
a. for all values of y
d. cannot determine
37.
y
y
2x
1000x
log 6 36x
log 1000 x
x
ey = x
d.
6x
?
c. 10x
- 12 -
d.
?
c. 2 x
b. 36x
b. 3
?
c. x = e
Which expression is equivalent to
a.
c. for some values of y
y
b. e = x
Which expression is equivalent to
a.
39.
b. for no values of y
Which expression is equivalent to ln x = y
a. 10 = x
38.
?
d. 3x
Unit 8 Worksheet 5
On pg. 507 in our text are the Laws of Logarithms
log b MN =
1.
Multiplication Property:
log b M + log b N
M
= log b M − log b N
N
3.
Power to a Power Property:
log b M n = n log b M
log10 6 = .7782 , use the Laws and the given
If you are given: log10 4 = .6021 and
2.
log b
Quotient Property:
to find the following. Justify each step with the properties listed above or basic operations property.
Example
log10 24
log10 (4 ⋅ 6)
Factors of 24
log10 4 + log10 6 Multiplication Property
.6021 + .7782
Substitution Property
1.3803
Addition
2.
log10 16
3.
6.
log10 6
7.
log10
3
2
log10
4.
1
4
log10 2 (hint: 2 = 4 )
5.
log10 36
8.
1
log10 ( )
16
Even though we were only given log10 4 and log10 6 we know log10 10 = 1 and log10 100 = 2
9.
log10 40
10. log10 400
In the preceding problems we had to work with decimal values. The following problems involve the
same 3 laws of logarithms, but we will use variables instead of decimals.
Given:
log 2 9 = c
Find the following in terms of
and
log 2 10 = d
c and d
11.
log 2 90 =
12.
log 2 81 =
13.
10
log 2 ( ) =
9
14.
log 2 10 =
15.
1
log 2 ( ) =
9
16.
1
log 2 ( )
10
17.
log 2 3
18.
log 2 900 =
19.
log 2 ( 3 9) =
20.
You were given the log 2 9 and log 2 10 , but you also know log 2 2 = 1, use this to find
log 2 18 =
- 13 -
Select the correct multiple choice:
21.
22.
23.
log xy2 =
a) 2 log xy
b) 2 log x + log y
c) 2 log x + 2 log y
log x • log y =
a) log (x + y)
b) log (x • y)
c) log x + log y
x
y
b)
b) 6x
log
1
•
2
d) 16x
c. 2 + x
d. 2x
3
2
c.
1
log
2
3
d.
1
log 3
2
log x + log y + log z =
b. log (x • y • z)
c. log x • log y • log z
log x (x w ) =
a. log w
29.
c) 8x
b. log 2 • log x
3 ) b. log
a. log (x + y + z)
28.
d) neither ‘a’ or ‘b’
3 =
a. log (
27.
c) both ‘a’ and ‘b’
log 2x =
a. log 2 + log x
26.
log x
log y
log 1004x =
a) 4x
25.
d) none of these
log x – log y =
a) log
24.
d) log x + 2 log y
b. log x w
c. w
Which student solved for x correctly in the following problem?
Alice
d. x w
2 log x = 4
Bob
Carl
2 log x = 4
2 log x = 4
2 log x = 4
log x2 = 4
log x2 = 4
log x2 = 4
log x2 = 4
x2 = 4
x2 = 4
x2 = 104
x = 2
x= ± 2
x2 = 10000
2 log x = 4
x = 100
- 14 -
David
x2 = 104
x2 = 10000
x = ± 100
30.
Which student solved for x correctly in the following problem?
Astro
2 log 3 + log x = log 36
Bella
2 log 3 + log x = log 36
2 log 3 + log x = log 36
log 9 + log x = log 36
log 9 + log x = log 36
log 9x = log 36
log (9 + x) = log 36
9x = 36
9 + x = 36
x= 4
x = 27
Chu
2 log 3 + log x = log 36
Domingo
2 log 3 + log x = log 36
2(log 3 + log x) = log 36
2(log 3 + log x) = log 36
2 log 3x
log 3x
= log 36
2
2 log 3x
= log 36
2
= log 36
log (3x)
= log 36
3x2
=
36
9x2
=
36
x2
=
12
x2
=
4
x
=
12
x
=
2
Unit 8 Worksheet 6
2x = 10
A. If we write log 2 10 in exponential form we get
approximate the value of this.
We know
23
2x
24
We are going to have to
=
8
=
10
=
16
So the exponent, x, will be between the consecutive integers 3 and 4.
B.
log3 25 becomes 3x = 25
32 = 9
3 x = 25
3 3 = 27
So x is between 2 and 3.
Between what 2 consecutive integers will x lie?
Would it be closer to 2 or closer to 3?
______
Determine which two integers the following logarithms lie between:
1.
log 2 30
2.
log 7 9
4.
log3 200
5.
log10 7500
3.
log4 100
You can convert all logarithm problems to equivalent logarithms with base 10 or e. Below
is the formula to convert logarithms to any base.
Change of Base
- 15 -
log c a
log c a
is currently in base ‘c’. To change it, write it as a fraction
=
log a
log c
You’ll notice that no base was given. You can use any base. For example:
log c a
=
log a
log c
log 6 a
log 6 c
=
or
log 4 a
log 4 c
or
log 8 a
log 8 c
Change of Base Formula
log c a
=
log b a
(where ‘b’ can be any positive base
log b c
≠ 1)
Since most calculators only work in base 10 or base e, it is best to change to one
of them.
log c a
=
log 10 a
ln a
ln c
or
log 10 c
Rewrite the following using the change of base formula. Change into the indicated base.
6.
log 5 7 to base 2
7.
log 9 4 to base 6
8.
log 2 3 to base 10
9.
log 8 5 to base e
You can use the change of base formula in reverse.
If you are given
log b a
log b c
log b a
log b c
you can condense it to a single log by dropping the base b.
=
log c a
Express the following as a single log:
- 16 -
10.
log 5 8
11.
log 5 7
log 9 12
log 2 6
12.
log 9 4
13.
log 2 10
log 11
14.
log 5
ln 4
ln3
Express the following as a single log. Then simplify the final answer.
15.
log 4 49
16.
log 4 7
18.
log 5 2
log 5 8
21.
log 5 7 =
19.
a. log 5 – log 7
22.
a.
log 8 81
17.
log 8 3
log 2
log 2
log 64
log 4
20.
b. log 7 – log 5
ln 32
ln 2
c. 7 • log 5
d.
log7
log5
log 8 20 =
log 3 20
log 3 8
23.
log 7 16
=
log 7 8
a.
log 716 – log 7 8
b.
 20 
log 

 8 
c.
log 20 – log 8
d.
b.
log 8 16
c.
log 2
2
d.
20 log 8
Algebra 2 Unit 8 Worksheet 7
Solve for x using common bases.
1
1.
3x=
2.
27
82+ x = 2
3.
4 1− x = 8
4.
27 2 x − 1 = 3
5.
4 3 x + 5 = 16 x + 1
6.
3 −( x + 5) = 9 4 x
7.
25 2 x = 5 x + 6
8.
6 x + 1 = 36 x − 1
9.
10 x − 1 = 100 4 − x
10.
5 x = 125
11.
49 x − 2 = 7 7
12.
6 x = 36 6
Solve for x using inverse properties of exponents.
13.
16.
19.
1
3
x =5
3
4
4 x = 108
5
3
( x + 5) − 2 =
30
14.
17.
20.
3
2
x =8
1
4
3x = 6
1
2
( x − 1) =
10
- 17 -
15.
18.
5
2
x = 32
5x
−
3
2
= 40
Unit 8 Worksheet 8
Solve for x. Some problems may have no solution.
1. log 2 x = 3
2. log 2 x = – 4
3. log 5 x = 3
4. log 2 (–2) = x
5. log x 144 = 2
6.
7. log 4 x =
1
2
8. log 8 x =
10.
log 1 6 = x
13.
log x 27 =
3
2
2
3
9.
5
log 5 23
= x
log 8 1 = x
11.
log 6 6 3 = x
12.
log 4 x = −
14.
log 7 (–49) = x
15.
log ( −9) x =
3
2
1
2
16. log16 x = –
1
2
17.
log 7 0 = x
18.
log 5 0 = x
19. log1 x = −
1
2
20.
log x 8 = – 1
21.
log x 16 = 2
22. log 3 (27 3) = x
23.
log 10 5 = x
24.
log x 8 =
25. log 5 (25 3) = x
26.
log 2 (4 5) = x
27.
log 2 7x = log 2 98
28. 3 log 5 4 = log 5 2x
29.
30.
2 ln 9 = ln 3x
9
log 7
x
= log 7 5
4
3
4
31.
log 7 2 x = log 7 16
32.
log 5 (2x + 12) = log 5 (3x + 4)
33.
2 log 8 x = log 8 100
34.
log 8 3 2 x = log 8 81
- 18 -
Algebra 2 Unit 8 Worksheet 9
Solve for x using properties of logs. On problems involving
terms of
π
π
or e leave answers in
or e. Do not approximate. Some problems will have no solution.
1.
log 7 x = log 7 2 + log 7 3
2.
log 6 x = 2 log 6 3 + log 6 5
3.
log 5 (x + 3) = log 5 8 – log 5 2
4.
log x – log (x – 5) = log 6
5.
ln (3x + 5) – ln (x – 5) = ln 8
6.
log 11 x =
7.
log 5 2 x = log 125
8.
log 6 9 + log 6 x = 2
9.
log x + log 25 = 3
10.
log 2 52 – log 2 x = 2
11.
2 log 6 2 + log 6 18x = 3
12.
ln 4 x = ln 8
13.
log π x
14.
log π 5 + log π x =
7
15.
log 64 32 = x
16.
log 6 x + log 6 (x – 5) = 2
17.
2 log 4 x = 3
18.
ln x = 2
19.
ln x + ln 5 = 4
20.
ln x – ln 6 = 2
21.
log 2 4x – log 2 (x – 1) = 3
22.
log 2 x + log 2 (x – 6) = 4
23.
2 log 2 + log x = 2
24.
2 ln 7 + ln x = 4
25.
log 20 + log 5 = x
26.
log 6 9 + log 6 4 = x
27.
log 5 (2x – 7) = 0
28.
ln (x – 9) = 1
29.
Identify which step has the error in the solution of 2 log 7 x = log 7 2 + log 7 50
= 3
Step 1:
2 log 7 x = log 7 (2 • 50)
Step 2:
2 log 7 x = log 7 100
 100 
log 7 

 2 
Step 3:
log 7 x =
Step 4:
log 7 x = log 7 50
Step 5:
x =
50
- 19 -
3
log 11 9 + log 11 2
2
30.
Which line has an error in it?
log 6 6 + log 6
31.
1.
log 6 6 6
2.
6x = 6 6
3.
6 x = 6 1 •62
= x
1
4.
x
6 =6
1
2
5.
x=
1
2
What multiple choice helps when solving
a. 32 ÷ 2 = 16
32.
6 = x
b. 2 • 32 = 64
c. p log x = log x
33.
p
5
1
d. 2 = 2
log 5 x + log 5 4 = log 5 24
b. log x + log y = log (xy)
d. log x – log y = log
What multiple choice helps when solving
a. ln x = ln e x
?
c. 32 = 2
What multiple choice helps when solving
a. log x + log y = log (x + y)
2 x = 32
b. e ≈ 2.718
- 20 -
x
y
ln x = 4
c.
41 = 4
d. 4 0 = 1
Unit 8 Worksheet 10
CALCULATORS ARE NOT ALLOWED
If we are given log2 2x = log2 5 , how would we solve for the exponent, x?
We use logarithms to help us solve these exponential functions.
Equation: 2x = 5
log2 2x = log2 5
x = log2 5
(our calculator could give us a decimal approximation, but for now this is
how we write our answers)
Solve the following problems for x by introducing logs. Leave answers in log form.
1.
7 x = 12
2.
5 x = 30
3.
10 x = 92
4.
8 2x = 74
5.
4 x+3 = 22
6.
e x = 43
Choose the correct multiple choice response:
7.
a.
8.
a.
7 x = 14
x=2
If x = log 4 15
x<0
9.
10 x = 200
a.
x = log 200
10.
ex = 4
a.
x = log 4
11.
2 x + 1 = 13
a.
x = log 2 13 − 1
b.
x = log 14
c.
x=
log14
log 7
d.
x = log 2
which is true about x?
b.
0<x<1
c.
1<x<2
d.
x>2
b. x = log 200 10
c.
x = 20
d.
x = 10
b.
x = ln 4
c.
x = ln e 4
d.
x= 4
b.
x= 6
c.
x=
d.
x = log 6
- 21 -
log12
log2
12.
Which step has the error:
ln 8 + ln x = 5
Step 1
ln 8x = 5
Step 2
8x = 10 5
Step 3
8x = 100,000
Step 4
13.
x =
Which step has the error:
100, 000
8
7 x+1 = 9
Step 1
log 7 7 x+1 = log 7 9
Step 2
x + 1 = log 7 9
Step 3
x
= log 7 9 – 1
Step 4
x
=
log 7 8
Algebra 2 Unit 8 Review 1
CALCULATORS ARE NOT ALLOWED
Simplify:
1.
125
1
3
2.
100
1
−
2
Write the following in logarithmic form.
6.
5.
4 3 = 64
10 −1 = 0.1
3
−
4
3.
16
7.
e 1 = 2.718
4.
 1 
 
 32 
8.
Write the following in exponential form.
 1 
9.
log 2 16 = 4
10.
log 5   = − 2
 25 
11.
log 1000 = 3
12.
14.
logπ 31 = 3
ln 148 = 5
13.
log 7 1 = 0
Simplify. Some problems will have no answer.
log 8
15.
16.
log 4 64
5 5
 1 
18.
log 5 0
19.
log 7  
 49 
1
21.
22.
log 2 8
ln  
e
- 22 -
17.
ln (e 2 )
20.
23.
ln (1)
log 8 ( 3 2)
−3
5
ab = c
24.
log  1  9
25.
 −1 
log 9  
 3 
1
log  
 10 
28.
ln (e 7 ) 29.
 
 3
27.
log 7 7 2 x
26.
log 5 1
30.
log 6 36 4 x
Solve for x.
On problems involving π or e leave answers in terms of π or e . (Do not approximate.)
Some problems will have no solution. Some problems will have answers in log terms.
1
4x
3–x
x–3
x
31.
3 = 3
32.
4 = 23
33.
2
= 16
34.
6
37.
x
= 11
x
3
2
36.
x = 64
39.
2 (7 x −1) 3 − 4 =
0
42.
e x +1 = 30
1
log x   = − 1
2
45.
log x 125 =
47.
log 5 (– 5) = x
48.
log π π 3 = x
50.
ln x = – 2
35.
5 =
2x2 =6
38.
9 2 x = 17
40.
e x = 23
41.
9
43.
log 5 x = – 3
44.
46.
ln x = 7
49.
1
ln  4  = x
e 
51.
log 6 4 + log 6 x = 2
52.
log 6 4 + log 6 x = log 6 12
53.
log 7 (x + 3) – log 7 x = log 7 2
54.
ln (x) + ln (3) = ln (x + 4)
55.
log x + log (x – 3) = 1
56.
2 log 6 x + log 6 3 = log 6 75
125
1
1
2x
= 27
x–1
Express as a single log and simplify, if possible.
57.
log 5 10 + log 5 4
58.
log 672 – log 6 2
60.
63.
log 7 11
log 7 4
log 6 8
log 6 2
Given:
log 2 = .3010
66.
log 12
70.
log
72.
log 72
67.
59.
3
4
2 log 210 – log 2 25
61.
log 50 + log 4 – log 2 62.
1
1
log 27 − log 9
3
2
64.
ln18
ln 5
65.
log 6 = .7781
log 3
1
2
(this is equivalent to log )
3
6
68.
Find the following:
log 4
71.
- 23 -
log 32
log 5
69.
log 20
log
1
2
Given:
73.
log 3 = k
log 15
log 5 = f
74.
1
log  
5
78.
1
y=  
3
Find the following:
75.
log 50
79.
y = log 4 x
76.
log 45
Graph the following:
77.
y= 7
x
x
Answer individual questions:
80.
Between what 2 consecutive integers does
81.
If the equation y = 4 x is graphed, which of the following multiple choice
values of x would produce a point closest to the x-axis?
a.
82.
log 1230 lie?
1
4
b.
0
c.
–2
d.
3
A radioactive substance decays by the given formula. How much of a 160 gram
sample will remain after 6 hours?
t
 13
y= A  
2
A = initial amount
t = time in hours
83.
A radioactive element decays over time as shown in the table below.
Which multiple choice equation expresses the amount of grams, y, present at
hour, h?
84.
Given the equation
a.
85.
x<0
b.
log 4
b.
b.
b.
log 3 24
log 3 6
h
h
x< 0
log 15
Which multiple choice is equivalent to
a. log 4
1
y = 100  
2
1
d.
y = 100
c.
y = •h • g
2
y = log x, which multiple choice statement is valid?
Which multiple choice is equivalent to
a.
86.
a.
hour grams
0
100
6
50
12
25
1
g
y=
2
c.
x=0
d.
x>0
log 20 – log 5
c.
log 20
d.
log 5
log 100
log 6 24 ?
c. log 6 11 + log 6 13
- 24 -
d. (log 6 2)(log 612)
 1 6
 
2
87.
Which multiple choice is the solution to the equation 9x = 45 ?
a. x =
88.
89.
90.
log 45
log 9
b. x = 5
c. x = log 5
d. x = log 45 – log 9
Given the expression xn where x > 1 and n > 1, which multiple choice statement is true?
a. the value of xn = 0
b. the value of xn > 0
c. the value of xn < 0
d. the value of xn = 1
Given the expression xn where x > 1 and n =0, which multiple choice statement is true?
a. the value of xn = 0
b. the value of xn > 0
c. the value of xn < 0
d. the value of xn = 1
Given the equation y = xn where 0 < x < 1 and n > 1, which multiple choice statement is true?
a. y = 0
b. 0 < y < 1
c. y < 0
d. y = 1
Algebra 2 Unit 8 Review 2
CALCULATORS ARE NOT ALLOWED
Choose the correct multiple choice response in # 1 – 22.
1.
Write 73 = 343 in logarithmic form.
a) log 7 343 = 3
b) log 3 343 = 7
c) log 7 3 = 343
2.
Write log10 0.0001 = −4 in exponential form
d) log 3 7 = 343
6.
d)
10−4 = 0.0001
1
Evaluate log16 4
a) 2
b) −2
c)
d)
2
Solve for x: log x 9 = 2 a) 3 b) 4.5
c) −3, 3
d)
1
Solve for x: log 5 x = −3 a) −15 b) −125 c)
d)
125
Evaluate:
b) 6 c) 25
d) 36
log 5 56 a) 5
7.
Evaluate:
7 log7 49
8.
Solve:
log 2 (−8) =
x a) 3 b) −3
9.
Solve: log 2 y =
10.
Solve:
a)
3.
4.
5.
11.
12.
13.
0.0001−4 = 10
b)
−410 =
0.0001
a) 7
1
log 2 125
3
c)
b) 2 c) 1
a) 5
b) 2
d) 49
c)
c) 375
1
3
−1
3
125
3
d)
d)
100.0001 =
−1
2
81
−1
125
e)
−4
e)
e)
e)
1
4
−3
3
5
none of these
e) none of these
e) none of these
e) none of these
1
b) 4 c) 16 d) 32 e) none of these
=
log 2 x
4 log 2 2 − log 2 4 a) 2
2
Solve: log 4 (m − 1) + log 4 (m − 1) =
2 a) 3 b) 5 c) 9 d) −3, 5 e) none of these
Solve: log 2 (x) + log 2 (x + 2) = 3
a) 2 b) 4 c) - 4
d) 2, – 4 e) 2, 4
Solve: log x 9 = 2
a) 3
b) - 3
c) 3, -3
d)
3 e) none of these
- 25 -
14.
15.
16.
17.
18.
19.
Given: log 2 = c and log 7 = d , Find: log 56 a) c 3 + d b) c3d
d) 3c + d
e) none of these
1
−1
1
Solve for x: log x ( ) = −3 a)
b) 2
c)
d)
2
8
2
Solve for x: log 5 0 = x a) 0
b) 1
c) 5 d) −1
e)
1
Solve for x: 2 log 5 6 − log 5 27 =
log 5 x a) 33 b) 2 c) 3 d)
3
1
1
b) n −1 c) − n d) 1
If log 4 7 = n, find log 4 ( ) a)
7
n
If log 2 = c
and log 3 = d,
find log 6
a)
20.
1
1
c+ d
2
2
22.
c)
d)
cd
c+d
−2 e)
none of these
none of these
4 e) none of these
− n e) none of these
1
e)
cd 2 f) none of these
Which of the following is true about the graph of y = log x ?
a)
it passes through (0,1)
b)
it lies in quadrants 1 and 2 only
c)
21.
1
c+d
2
b)
c) c 3 + d
it is a decreasing graph
d)
the value of x will never be 0
3log 7 4 + 2 log 7 2 = log 7 x
b)
16
c) 256
4
Find the value of x:
= log125 x
3
a)
50
b)
5
c)
25
d)
Solve for x:
a)
64
d)
32
625
True or False
23.
25.
27.
10log x = x
log 3 9 = log 7 49
24.
eln x = x
26. 22 x = 4 x
1
23+3 x = 81+ x
28.
2x 2 =
2x
Simplify:
29.
log 5 + 2 log 4 – 3 log 2
30.
log 7 32
log 7 2
31.
log 6 3 + log 6 12
34.
log 6 3 + log 6 x = log 62
Solve for x:
x
32.
1
3
  = 4
2
35.
log8 x =
38.
1 1
  =
2 8
−1
3
33.
log 6 (x + 5) + log 6 x = 2
1
36.
1
8
log 3 (27 3) = x
37.
x2 =
39.
40.
log 3 (– 9) = x
x
ln x = 4
- 26 -
x
43.
1
Graph: y =  
5
Express as a single log:
44.
Express as a single log and simplify: 2 log 6 2 + 2 log 6 3
45.
If log 5 2 = k
46.
Given the equation y = xn where x > 1 and n < 0, which multiple choice statement is true?
a) y = 0
b) y > 0
c) y < 0
d) y = 1
47.
If the equation y = 4x is graphed, which value of x would produce a point closest to the x axis?
2
1
a) 3
b) – 5
c) −
d)
3
3
x
1
If the equation, y =   is graphed, which value of x would produce a point closest to the
3
x-axis?
a) 7
b) 0
c) 2
d) – 6
41.
48.
42.
Graph:
y = log 2 x
2 ln 7 + ln 2
and log 5 3 = m
find log 5 30 in terms of k and m
x
9
49.
If the equation, y =   is graphed, which of the following values of x would produce a point
7
closest to the x axis?
3
4
1
2
a) −
b)
c)
d) −
5
7
3
3
50.
If the equation, x = log2y is graphed, which of the following values of x would produce a point
farthest from the x axis?
a) – 8
b) 2
c) – 3
d) 9
Simplify
51.
52.
53.
log327x
log416x
log82x
Unit 8 Review #3
1. Given the equation y = x n where x > 1 and n > 0 , which statement is true?
a) y = 0
b) y < 0
c) y > 1
d) 0 < y < 1 e) y is undefined
2. Given the equation y = x n where 0 < x < 1 and n > 0 , which statement is true?
a) y = 0
b) y < 0
c) y > 1
d) 0 < y < 1 e) y is undefined
3. Given the equation
y = x n where 0 < x < 1 and n < 0 , which statement is true?
- 27 -
a) y = 0
b) y < 0
c) y > 1
d) 0 < y < 1
e) y is undefined
4. Bacteria are growing exponentially with time as shown in the table below. Write the equation that
expresses the number of bacteria, y, present at any time, t ?
Bacteria Growth
Hour
Bacteria
0
5
1
10
2
20
5. Bacteria are decaying exponentially with time as shown in the table below. Write the equation that
expresses the number of bacteria, y, present at any time, t ?
Bacteria Growth
Hour
Bacteria
0
100
1
50
2
25
Simplify the following:
1
8. log 3  
9
6. log 3 ( −9 )
7. log ( −3) 9
9. log 3 9 x
10. log 5 125 x
Approximate the following:
11. log 2 9
12. log 4 3
13. log 3 30
- 28 -