Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED

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

wand

x

logxw_,

log 1

b , 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 1 – 33

10. Solve exponential equations without a common base. Worksheet 10 1-13

Review

Review Worksheet 1 1 – 90 Review Worksheet 2 1 – 53 Review Worksheet 3 1 - 13

Algebra 2 Unit 8 Worksheet 1

CALCULATORS ARE NOT ALLOWED

Simplify: 1.

3

4 2.

2

−3 3. 2

1

5

 

 

 

4. 1 2

1

4

−

 

 

 

5.

7

0 6.

1

4 7. 4

1

2

−

 

 

 

8. 2

9

− 9. 5 -2 _10. 1 2

36

11. 2 3

64

12. 1 2

16

− Definition:

The function defined by y =

b

x is called an exponential function with base b Requirements: b > 0, b

≠

1

Characteristics of exponential functions:

The basic graph of an exponential function looks like the following:

An increasing exponential if they rise as they go from left to right. 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. 13.

y = 2

x 14. y =

3

x

15.

1

2

 

=  

_{ }

x

y

16.

1

5

 

=  

_{ }

x

y

x y 2 1 0 –1 –2 Increasing Decreasing x y 2 1 0 –1 –2 x y 2 1 0 –1 –2 x y 2 1 0 –1 –2

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 18.

1

3

x

y  

_{=  }

 

19.

y = 5

x _20.

1

4

x

y  

_{=  }

 

Create a table of values in problems 21 -23 and then graph on graph paper.

21. y = 1 x

22. y = 0x 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. If the equation of y =

5

x is graphed, which of the following values of x would produce a point closest to the x-axis?

a. 0 b. –1 c. 2 3 d. 74 26. If the equation of y =

1

2

x

 

 

 

is graphed, which of the following values of x would

produce a point closest to the x-axis? a. 1₄ b. 3₄ c. 5₃ d. 8₃ 27. If the equation of y =

1

3

x

 

 

 

is graphed, which of the following values of x would

produce a point closest to the x-axis? a. 0 b. –1 c. 2

3 d. 74

28. Which multiple choice ordered pair represents the y-intercept for the function y = 2 x _?

a. (0,0) b. (0, 1) c. (0, 2) d. there is no y-intercept 29. Select the correct multiple choice response.

The graph of y = 5 x lies in which quadrants?

30. Select the correct multiple choice response. The graph of y = 1 10 x    

  contains which of these points? a. (0, 0) b. (0, 10) c. (0, 1) d. (0, 1

10)

31. Which multiple choice ordered pair represents the x-intercept for the function y = 4 x _?

a. (0, 0) b. (0, 1) c. (1, 0) d. there is no x-intercept

32. Use the graph of y =

2

x to answer the following multiple choice question. If the equation y =

2

x is graphed, which of the following values of x would produce a point closest to the x-axis?

a. 1

4 b. 34 c. 53 d. 83

33. Given the expression

x

n where x > 1 and n > 1, which multiple choice statement is true?

a. the value of

x

n = 0 b. the value of

x

n > 0 c. the value of

x

n < 0 d. the value of

x

n = 1

34. Given the expression

x

n

where x > 1 and n = 0, which multiple choice statement is true?

a. the value of

x

n = 0 b. the value of

x

n > 0 c. the value of

x

n < 0 d. the value of

x

n = 1

35. Given the equation y =

x

n

where 0 < x < 1 and n > 1, which multiple choice statement is true?

a. y = 0 b. y > 0 c. y < 0 d. y = 1

36. Given the equation y =

x

n

where x > 1 and n < 0, which multiple choice statement is true?

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

Exponential Decay.

Such a situation is called

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.

For the following word problems we will be using the exponential equation y = A

( )

b . ht

Where A is the initial amount

b is the amount of growth (or decay) that occurs in h time t is time

1. 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?

Use the formula y = A

t 6 1 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? Use the formula N = N0

t h 1 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. The radioactive gas radon has a half-life of 3 days. How much of an 80 gram sample will remain after 9 days?

5. The radioactive gas radon has a half-life of approximately 31

2 days. About how much of a 200 gram sample will remain after 1 week?

6. 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₎

60 t

A = initial population t = time elapsed in years

7. 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?

8. 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?

9. 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.)

10. The growth of a town doubles every year. If there are 64,000 people after 4 years, find the initial population.

11. 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 ?

a. N = 400 x b. N = 400 + 2x c. N = 800 • 2–x d. N = 400 • 2x

12. 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. y = 30,000 •2t b. y = 10000 • 3t

c. y = 3 • 10t d. y = 30,000 + 2t

13. An epidemic of bubonic plague grew exponentially by the formula

A = A

0

• 2

t 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.

Use estimation for the following multiple choice questions:

Choose the best multiple choice response for the following: 14. 9 4

3

a. 1.3 b. 3.9 c. 11.8 d. 35.5 15.

(

) ( )

3 2

12 2

•

a. 16.9 b. 33.9 c. 67.3 d. 117.5

16. A radioactive element decays over time according to the equation:

y = A

1

300

2

 

 

 

t

If 1000 grams were present initially, how may grams will remain after 650 years?

a. 444 b. 222 c. 111 d. 55.5

17. Boogonium decays using the formula: A = I • t h

2

−

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 • t h

2

−

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

Logarithms (or logs) are used to find the exponents to help us solve

exponential equations.

Structure of a logarithm:

log

_b

y

=

x

b is the base

y is the value

x is the exponent on b to

yield y

Example: Simplify

log 8

₂

=

?

=3 (because 2

3

_{= 8)}

Simplify #10-29.

10. log6 36 11. log2 16 12. log10 100 13. log3  ₉1

  14. log 2 2 2 15. log7 1 16. log5 125 17. log4 16

18. log3 81 19. log6 6 20. log3 1 21. log8 4 22. log5₂₅1    23. 2 1 log 8       24. log 6 6 6 25. log 25 55 26. 3 7

log

49

27. 5 3

log 9

28.

log

₂ 3

1

4













29. 10

1

log

100













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)

30. log 100 31.

log

1

10













32. log 1 33. log 10

Unit 8 Worksheet 4

Log Rules :

(

b and y must be positive numbers, b

≠

1

)

log

b

y x

=

b

x

=

y

logbx

b

=

x

log

_b

b

y

= y

log 1

b

= 0

π ≈

3.14

e ≈

2.718

Remember, if no base is shown assume it is base 10, the common log.

log y x

=

log y x

₁₀

=

10

x

= y

If base e is used it is called a natural log. Instead of writing log we use ln

ln y x

=

log

_e

y x

=

e

x

= 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

log (of a negative number)

With bases, you can’t do

log

_{0 base}

or

log

_{1 base}

or

log

_{negative base} Verify the log by rewriting the equation into exponential form.

1. log2 32 = 5 2. log3 9 = 2 3. log7 7 = 1

2 4. log3 181 = – 4 Rewrite the equation in logarithmic form.

5. 43 _{= 64 6.}

9

32 _{= 27 7.}

10

−2

=

0 01

_.

_8. 3 4

1

16

8

−

=

Simplify:

9.

5

log 235 10.

log 2

₂ 7 11.

10

log9 12.

log 1

₁₂ 13.

log 49

₇ 14.

log 64

₈ x 15.

log 16

₄ x 16.

log 16

₂ x 17.

log 1

₈ 18.

log 7

₇ 5 19.

3

log 113 20.

log 6

₆

Write each equation in exponential form.

21. ln 8 = 2.08 22. ln 100 = 4.61 23.

log 9.86 2

π

=

24.

log1000 3

=

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 x

₂ 27. y =

log x

₅ 28. y = ₁ 4

log x

29. Graph y =

3

x and y =

log x

₃ on the same grid.

Choose the correct multiple choice.

30. Which is equivalent to

16

12 _{= 4 ?} a.

 

_{ }

 

4

1

log

2

= 16 b. 16

1

log

2

 

 

 

= 4 c. 16

1

log 4

2

=

d. log 4 16 = 1 2 31. Which is equivalent to

log

_m

n p

=

?

a.

m

n

= p

b.

m

p

= n

c.

n

p

= m

d.

p

n

= m

32

.

Which is equivalent to

log k = w

?

a.

10

w

= k

b.

1

w

= k

c.

k

w

= 10

d.

10

k

= w

33. Given:

y = 5

x 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

34. When is the following statement true?

7

log7x _{= x}

a. for all values of x b. for some values of x c. for no values of x d. can’t determine

35. In the equation

log

_x

y z

=

which statement is true about the value of z ? a. z must always be positive 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

₆ y = y ?

a. for all values of y b. for no values of y c. for some values of y d. cannot determine

37. Which expression is equivalent to

ln x =

y ?

a.

10

y

= x

b.

e

y

= x

c.

x

y

= e

d

. e

y

= x

38. Which expression is equivalent to

log 36

₆ x ?

a. 2x b. 36x c. 2 x _{d. 6} x

39. Which expression is equivalent to

log 1000

x ?

Unit 8 Worksheet 5

On pg. 507 in our text are the Laws of Logarithms

1. Multiplication Property:

log

_b

MN

=

log

_b

M

+

log

_b

N

2. Quotient Property:

log

_b

M

N

=

log

b

M

−

log

b

N

3. Power to a Power Property:

log

_b

M

n = n

log

_b

M

If you are given:

log 4

₁₀ =

.6021

and

log 6

₁₀ =

.7782

, use the Laws and the given to find the following. Justify each step with the properties listed above or basic operations property. Example

log 24

₁₀

10

log (4 6)

⋅

Factors of 24 10

log 4

+

log 6

₁₀ Multiplication Property .6021 + .7782 Substitution Property 1.3803 Addition 2.

log 16

₁₀ 3.

log

₁₀

3

2

4. 10

1

log

4

5.

log 36

10

6.

log

₁₀

6

7.

log 2

₁₀ (hint: 2 =

4

) 8.

log (

₁₀

1

)

16

Even though we were only given

log 4

₁₀ and

log 6

₁₀ we know

log 10

₁₀ = 1 and

log 100

₁₀ = 2

9.

log 40

₁₀ 10.

log 400

₁₀

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 9

₂ =

c

and

log 10

₂ =

d

Find the following in terms of

c

and

d

11.

log 90

₂ = 12.

log 81

₂ = 13.

log (

₂

10

)

9

= 14.

log

₂

10

= 15.

log ( )

₂

1

9

= 16. 2

1

log (

)

10

17.

log 3

₂ 18.

log 900

₂ = 19.

log ( 9)

₂ 3 =

20. You were given the

log 9

₂ and

log 10

₂ , but you also know

log 2

₂ = 1, use this to find 2

Select the correct multiple choice:

21. log xy2 =

a) 2 log xy b) 2 log x + log y c) 2 log x + 2 log y d) log x + 2 log y 22. log x • log y =

a) log (x + y) b) log (x • y) c) log x + log y d) none of these 23. log x – log y =

a) log x

y b)

logx

log y c) both ‘a’ and ‘b’ d) neither ‘a’ or ‘b’ 24. log 1004x =

a) 4x b) 6x c) 8x d) 16x

25. log 2x =

a. log 2 + log x b. log 2 • log x c. 2 + x d. 2x

26. log 3 = a. log (1 2 • 3 ) b. log 3 2 c. 1 2 log 3 d. 1 2 log 3

27. log x + log y + log z =

a. log (x + y + z) b. log (x • y • z) c. log x • log y • log z 28. log x (x w _{) =}

a. log w b. log x w c. w d. x w

29. Which student solved for x correctly in the following problem? 2 log x = 4 Alice Bob Carl David 2 log x = 4 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 x2 = 104 x = 2 x = ± 2 x2 = 10000 x2 = 10000

30. Which student solved for x correctly in the following problem? 2 log 3 + log x = log 36

Astro 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 Domingo

2 log 3 + log x = log 36 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 36 2 log 3x = log 36 log 3x2 = log 36 log (3x)2 = log 36 3x2 = 36 9x2 = 36 x2 = 12 x2 = 4 x = 12 x = 2

Unit 8 Worksheet 6

A. If we write

log 10

₂ in exponential form we get

2

x

=

10

We are going to have to approximate the value of this. We know

2

3 = 8

2

x = 10 4

2

= 16

So the exponent, x, will be between the consecutive integers 3 and 4.

B.

log 25

₃ becomes

3

x

=

25

Between what 2 consecutive integers will x lie?

3

2 _{= 9}

3

x _{= 25}

3

3 _{= 27}

So x is between 2 and 3. Would it be closer to 2 or closer to 3? ______ Determine which two integers the following logarithms lie between:

1.

log 30

2 2.

log 9

7 3.

log 100

4

4.

log 200

₃

5.

log 7500

₁₀

You can convert all logarithm problems to equivalent logarithms with base 10 or e. Below is the formula to convert logarithms to any base.

log

c

a

is currently in base ‘c’. To change it, write it as a fraction

log

c

a

=

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

= 6₆

log a

log c

or 4₄

log a

log c

or 8₈

log a

log c

Change of Base Formula

log

c

a

= b

b

log a

log c

(where ‘b’ can be any positive base

≠

1)

Since most calculators only work in base 10 or base e, it is best to change to one of them.

log

c

a

= 10 10

log a

log c

or

ln a

ln 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 b

b

log a

log c

you can condense it to a single log by dropping the base b.

b

b

log a

log c

=

log

c

a

10. 5 5 log 8 log 7 11. 9 9 log 12 log 4 12. 2 2 log 6 log 10 13. log 11 log 5 14. ln 4ln3 Express the following as a single log. Then simplify the final answer.

15. 4 4

log 49

log 7

16. 8 8

log 81

log 3

17.

log 64

log 4

18. 5 5

log 2

log 8

19.

log 2

log 2

20.

ln32

ln2

21. log 5 7 =

a. log 5 – log 7 b. log 7 – log 5 c. 7 • log 5 d. log7_log5 22.

log 20

₈ = a. 3 3

log 20

log 8

b.

20

log

8













c. log 20 – log 8 d. 20 log 8

23. 7

7

log 16

log 8

=

a. log 716 – log 7 8 b. log 8 16 c. log 2 d. 2

Algebra 2 Unit 8 Worksheet 7

Solve for x using common bases. 1. 3 x _{= 1} 27 2.

8

2 x + = 2 3.

4

1−x = 8 4.

27

2x −1 = 3 5.

4

3x+5

=

16

x+1 6.

3

−(x+5)

₌

9

4x 7.

25

2x

=

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. 1 3

5

x =

14. 3 2

8

x =

15. 5 2

32

x =

16. 3 4

4

x =

108

17. 1 4

3

x =

6

18. 3 2

5

x

−

=

40

19. 5 3

(

x +

5)

−

2 30

=

20. 1 2

(

x −

1)

=

10

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.

5

log 235 = x 7. log 4 x = 1

2 8. log 8 x = 23 9. log 8 1 = x 10. log 16 = x 11. log 6 6 3 _{= x 12. log 4 x = 3}

2 −

13. log x 27 = 3₂ 14. log 7 (–49) = x 15. log( 9)− x= 1₂

16. log16 x = – 1 2 17. log 7 0 = x 18. log 5 0 = x 19. ₁ 9

1

log

2

x = −

20. log x 8 = – 1 21. log x 16 = 2

22. log (27 3) = x 3 23. log 10 5 = x 24. log x 8 = 3₄

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. log 7

4

x _{= log 7 5 30. 2 ln 9 = ln 3x}

31.

log 2

₇ x

=

log 16

₇ 32. log 5 (2x + 12) = log 5 (3x + 4) 33. 2 log 8 x = log 8 100 34.

log 3

₈ 2x

₌

log 81

₈

Algebra 2 Unit 8 Worksheet 9

Solve for x using properties of logs. On problems involving

π

or e leave answers in terms of

π

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 = 3

2 log 11 9 + log 11 2 7.

log 5

2x

=

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 x

_π = 3 14.

log 5 log

_π

+

_π

x

=

7

15.

log 32 x

₆₄

=

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 Step 1: 2 log 7 x = log 7 (2 • 50)

Step 2: 2 log 7 x = log 7 100 Step 3: log 7 x =

log

₇

100

2













Step 4: log 7 x = log 7 50 Step 5: x = 50

30. Which line has an error in it? log 6 6 + log 6 6 = x 1. log 6 6 6 = x 2. 6x ₌6 6 3.

₆

x

₌

_{6 6}

1

_•

₂1 4. 6x ₌621 5. x = 1 2

31. What multiple choice helps when solving 2 x _{= 32 ?}

a. 32 ÷ 2 = 16 b. 2 • 32 = 64 c.

32 = 2

5

d.

2

1

= 2

32. What multiple choice helps when solving log 5 x + log 5 4 = log 5 24

a. log x + log y = log (x + y) b. log x + log y = log (xy) c. p log x = log x p _{d. log x – log y = log}

x

y

33. What multiple choice helps when solving ln x = 4

Unit 8 Worksheet 10

CALCULATORS ARE NOT ALLOWED

If we are given

log 2 log 5

2 x

=

2

, how would we solve for the exponent, x?

We use logarithms to help us solve these exponential functions.

Equation:

_{2 5}

x

=

log 2 log 5

₂ x

=

₂

x =

log 5

2

(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.

7

x

=

14

a. x = 2 b. x = log 14 c. x =

log14

log7

d. x = log 2

8. If x =

log 15

₄ which is true about x?

a. x < 0 b. 0 < x < 1 c. 1 < x < 2 d. x > 2 9.

10

x = 200 a. x = log 200 b. x =

log 10

_{200 c.} x = 20 d. x = 10 10.

e

x = 4 a. x = log 4 b. x = ln 4 c. x = ln e 4 _{d. x = 4} 11.

2

x

+ 1 = 13

a. x =

log 13 1

₂

−

b. x = 6 c. x =

log12

log2

d. x = log 6

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 x =

100, 000

8

13. Which step has the error:

7

x 1+

=

9

Step 1

log 7

₇ x 1+ ₌ 7

log 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. 1 3

125

2. 1 2

100

− 3. 3 4

16

− 4. 3 5

1

32

−













Write the following in logarithmic form.

5. 4 3 = 64 6. 1

10− =0.1 7. 1

2.718

e = 8. b

a = c

Write the following in exponential form.

9. log 2 16 = 4 10. log₅ 1 2 25  _{ = −}     11. log 1000 = 3 12. ln 148 = 5 13. log 7 1 = 0 14. log 31 3_π =

Simplify. Some problems will have no answer.

15. log 85 5 16. log 4 64 17. ln (e2) 18. log 5 0 19. 7 1 log 49       20. ln (1) 21. ln 1 e       22. log 28 23. 3 8 log ( 2)

24. ₁ 3 log_{ }9     25. log₉ 1 3 −       26. log 5 1 27. log 1 10       28. 7 ln (e ) 29. log 7₇ 2x 30. log 36₆ 4x 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.

31.

3

4x =

3

3 – x 32. 4 x = 23 33.

2

x – 3 = 1 16 34. 6 x = 11 35. 5x = 125 36. 3 2 ₆₄ x = 37. 1 2 2x = 6 38. 92x = 17 39. 1 3 2 (7x−1) − = 4 0 40. ex=23 41.

9

2x =

27

x – 1 42. ex+1 =30 43. log 5 x = – 3 44. log 1 1 2 x  = −    45. log x 125 = 3 4 46. ln x = 7 47. log 5 (– 5) = x 48. logππ3 = x 49. ln 1₄ e       = x 50. ln x = – 2

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

Express as a single log and simplify, if possible.

57. log 5 10 + log 5 4 58. log 672 – log 6 2 59. 2 log 210 – log 2 25

60. 7

7

log 11

log 4 61. log 50 + log 4 – log 2 62.

1 1 log 27 log 9 3 − 2 63. 6 6 log 8 log 2 64. ln18 ln 5 65. log 32 log 5 Given: log 2 = .3010 log 6 = .7781 Find the following:

66. log 12 67. log 3 68. log 4 69. log 1

2

70. log 1

3 (this is equivalent to log 2

6) 71. log 20

Given: log 3 = k log 5 = f Find the following: 73. log 15 74. log 1 5       75. log 50 76. log 45

Graph the following:

77. y = 7x 78. y = 1 3 x       79. y = log 4 x

Answer individual questions:

80. Between what 2 consecutive integers does log 1230 lie?

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. 1

4 b. 0 c. – 2 d. 3

82. A radioactive substance decays by the given formula. How much of a 160 gram sample will remain after 6 hours?

y = A 1 3 2 t       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 a. y = 1 2 g b. y = 100 1 2 h       hour, h? c. y = 1 2• •h g d. y = 100 6 1 2 h       84. Given the equation y = log x, which multiple choice statement is valid?

a. x < 0 b. x < 0 c. x = 0 d. x > 0 85. Which multiple choice is equivalent to log 20 – log 5

a. log 4 b. log 15 c. log 20

log 5 d. log 100

86. Which multiple choice is equivalent to log 6 24 ? a. log 4 b. 3

3

log 24

log 6 c. log 6 11 + log 6 13 d. (log 6 2)(log 612) hour grams

0 100

6 50

87. Which multiple choice is the solution to the equation 9x = 45 ? a. x = log 45

log 9 b. x = 5 c. x = log 5 d. x = log 45 – log 9

88. 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

89. 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

90. 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 3

7 =343 in logarithmic form.

a) log 343₇ = b) 3 log 343₃ = c) 7 log 3₇ =343 d) log 7₃ =343 2. Write log 0.0001₁₀ = − in exponential form 4

a) 0.0001−4 =10 b) −410 =0.0001 c) 10−4 =0.0001 d) 100.0001= −4 3. Evaluate log 4 ₁₆ a) 2 b) −2 c) 1 2 d) 1 2 − e) 1 4

4. Solve for x: log 9_x = a) 3 b) 4.5 2 c) −3, 3 d) 81 e) −3 5. Solve for x: log₅x= − a) 3 −15 b) −125 c) 1

125 d) 1 125 − e) 3 5 6. Evaluate: log 5₅ 6 a) 5 b) 6 c) 25 d) 36 e) none of these 7. Evaluate: ₇log 497 _{a) 7 b) 2 c) 1 d) 49 e) none of these}

8. Solve: log ( 8)₂ − = a) 3 b) −3 x c) 1 3 d) 1 3 − e) none of these 9. Solve: log₂ 1 log 125₂

3

y = a) 5 b) 2 c) 375 d) 125

3 e) none of these

10. Solve: 1 log₂ 4 log 2₂ log 4₂

2 x = − a) 2 b) 4 c) 16 d) 32 e) none of these

11. Solve: log (₄ m− +1) log (₄ m− = a) 3 b) 5 c) 9 d) 1) 2 −3, 5 e) none of these 12. Solve: log 2 (x) + log 2 (x + 2) = 3 a) 2 b) 4 c) - 4 d) 2, – 4 e) 2, 4 13. Solve: log x 9 = 2 a) 3 b) - 3 c) 3, -3 d) 3 e) none of these

14. Given: log 2=c and log 7=d, Find: log 56 a) c 3 + d b) c3d c) c 3 + d d) 3c + d e) none of these

15. Solve for x: log ( )1 3 8 x = − a) 1 2 b) 2 c) 1 2 − d) −2 e) none of these 16. Solve for x: log 0₅ = a) 0 x b) 1 c) 5 d) −1 e) none of these

17. Solve for x: 2 log 6₅ 1log 27₅ log₅

3 x − = a) 33 b) 2 c) 3 d) 4 e) none of these 18. If log 7 = n, find 4 4 1 log ( ) 7 a) 1 n b) 1 n− c) − n d) 1 − n e) none of these 19. If log 2=c and log 3 = d, find log 6

a) 1 1 2c+2d b) 1 2c+d c) cd d) c+ e) d 1 2 cd f) none of these

20. Which of the following is true about the graph of y=logx ?

a) it passes through (0,1) b) it lies in quadrants 1 and 2 only c) it is a decreasing graph d) the value of x will never be 0 21. Solve for x: 3log 4 + ₇ 2 log 2 = ₇ log x ₇

a) 64 b) 16 c) 256 d) 32

22. Find the value of x: 4 log₁₂₅

3= x a) 50 b) 5 c) 25 d) 625

True or False

23. 10logx =x 24. eln x =x 25. log 9₃ =log 49₇ 26. 22x = 4x 27. 23 3+ x = 1 8+x 28. 1 2 2x = 2x Simplify:

29. log 5 + 2 log 4 – 3 log 2 30. 7 7

log 32

log 2 31.

log

6

3 + log

6

12

Solve for x: 32. 1 2 x       = 3

4 33. log 6 (x + 5) + log 6 x = 2 34. log 6 3 + log 6 x = log 62

35. log₈ 1 3 x=− 36. log (27 3) = x ₃ 37. 1 2 1 8 x = 38. 1 1 2 8 x   =     39. ln x = 4 40. log 3 (– 9) = x

41. Graph: y = 1 5 x       42. Graph: y = log x2

43. Express as a single log: 2 ln 7 + ln 2

44. Express as a single log and simplify: 2 log 6 2 + 2 log 6 3

45. If log 5 2 = k and log 5 3 = m find log 5 30 in terms of k and m

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? a) 3 b) – 5 c) 2 3 − d) 1 3 48. If the equation, 1 3 x y=   

  is graphed, which value of x would produce a point closest to the

x-axis? a) 7 b) 0 c) 2 d) – 6 49. If the equation, 9 7 x y=   

  is graphed, which of the following values of x would produce a point

closest to the x axis? a) 3 5 − b) 4 7 c) 1 3 d) 2 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. log327x 52. log416x 53. log82x Unit 8 Review #3

1. Given the equation y=xn where x> and 1 n> , which statement is true? 0 a) y = 0 b) y < 0 c) y > 1 d) 0 < y < 1 e) y is undefined 2. Given the equation y=xn where 0< < and x 1 n> , which statement is true? 0

a) y = 0 b) y < 0 c) y > 1 d) 0 < y < 1 e) y is undefined 3. Given the equation y=xn where 0< < and x 1 n< , which statement is true? 0

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:

6. log₃

( )

− 9 7. log_{( 3)−} 9 8. log₃ 1

9      

9. log 9₃ x 10. log 125₅ x

Approximate the following: