Logarithms, Logarithmic Equations, Applications - proprep Logarithmic... · 3 3 log 12 10 2 2 x x...

College Algebra 2

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Logarithms, Logarithmic Equations,

Applications

Rules of Logarithms Questions:

1) Compute the value of the following logarithms:

5log 5 a. 125log 5 b.

1

2

log 16 c.


1

4

1log

8 a.

3

1

3

1log

9 b. 4

1

27

log 3 c.


3

5

1

25

log 125 a. 0.01 4

10log

1000 b.

4) Compute the value of the following expressions:

2 2log 10 log 6.4 a. 2 2log 768 log 6 b.

0.25 0.25log 80 log 5 c.


3 33log 6 log 3.375 a.

5 5 5 5log 50 log 20 log 2 log 4 b.


7 7 7

1log 81 2log 6 log 84

4 a. 3 3 32 2 2

1 1 3log 6 log 3 log 4

2 2 2 b.

College Algebra 2




3

3

log 16

log 8 a. 3

3 3

log 6 2

log 108 log 2

b.

8) Compute the value of the expression: 2 3log5 log50

1 log128 5log 2

Hint: convert whole numbers to logarithmic expressions using log n

an a

9) Compute the value of the following expressions (the logarithm is base 10):

log 27

log9 a.

log 24 log3

log 2

b.

1 log5

log 2 2log5

c.

10) Prove that the following equality holds (the logarithm is base 10): log9 2log5 log 4

2log10 log 2 log6

11) Given:

2log 7 a . Express the following in terms of a:

2log 14 a.

2log 49 b.

12) Given log4 a . Express the following in terms of a:

log16 a. log 2 b. log8 c.

13) Given: 3 36 log 5log , a b .Express the following in terms of a and b:

3log 30 a.

3log 1.2 b. 3log 150 c.

14) Compute the values of the following expressions, using the formula loga ba b :

2log 32 a. log210 b. 33log 43 c.


2log 38 a. 36log 4

6 b. 5log 64

3 5 c.


51 log 25

a. 31 log 23 27 b.

College Algebra 2



17) Compute the following expressions, using the formula log

loglog

ma

m

bb

a:

3 6log 6 log 3 a. 0.1 25log 5 log 100 b.

18) Compute the following expression, using the formula log

loglog

ma

m

bb

a:

81 32 7log 49 log 3 log 2

19) Prove the following equality,using the formula log

loglog

ma

m

bb

a:

3 5 3 2 3log 5 log 8 log 2 log 5 log 40

20) Given:

2log 5 a . Express the following in terms of a:

5log 2 a.

4log 5 b. 16log 5 c.

21) Given: log2 a . Express the following in terms of a:

log80 a. 8log 40 b.

80log 2000 c.

College Algebra 2



Final Answers:

1) a. 1x b. 1

3x c. 4x

2) a. 3

2x b. 6x c.

1

2x

3) a. 9

10x b.

1

8x

4) a. 6x b. 7x c. 2x

5) a. 6x b. 3x

6) a. 2x b. 8 2

7) a. 4

3 b. 1

8) 1

9) a. 1.5 b. 3 c. 1

10) proved

11) a. 1a b. 2a

12) a. 2a b. 1

2a c.

3

2a

13) a. a b b. a b c. 2a b

14) a. 3 b. 2 c. 64

15) a. 27 b. 2 c. 4

16) a. 10 b. 216

17) a. 1 b. -1

18) 1

10

19) Proved

20) a. 1

a b.

2

a c.

4

a

21) a. 3 1a b. 2 1

3

a

a

c.

3

3 1

a

a

College Algebra 2



Exponential Equations - Revisited Questions:

1) Solve the following equations:

2 5x a. 3 7x b.


12 5x a. 2 13 4 15x b.

Final Answers:

1) a. 2log 5x b.

3log 7x

2) a. 2log 5 1x b.

4log 5x

College Algebra 2



Logaritmic Equations - Type 1 Questions:


log 2 4 1 x x a. 2log 3 15 2 x x x b.


2

1

4

3log 4 11

2 x x a. 2

27

4log 8

3 x x b.


1 2

2

1log 3

4 11x x

a.

1log 2 1

2x x b.


1 2

2

log log 1 x x a. 12log log 5 3 2 x x b.


2

2 2log log 1 4 x a. 2

3log 1 4 x b.


7 9log 4 2log 0 x a. 1 2 3

2

log log 2log 1 x b.


2

1 2 5

2

log log log 10 14 1 x xa.

2

2 22 3log log log log 2 1 1 x x b.


ln 4x a. 2ln 4 1 0 x b.


ln ln 1x a. ln ln 1 0 x e b.

College Algebra 2




2log 3 17 6 x a. 2log 12 2 1 x x b.


3 1

23 3 log 12 10 2 xx a. 1

52 3 log 30 25 xx b.


5

1log 6 5 5

2 xx a. 1

2log 68 4 2 xx b.


3 3

2log 2 log 64

3x a. 5log 125 24 25 2 x x x b.


2 54log

4 3 4

xx a.

2 45log5 12

x x b.


3 12log3 4 x a.

24

1log 1

42 5

x

x b.

College Algebra 2



Final Answers:

1) a. 4x b. 5x

2) a. 1,3x b. 1,9x

3) a. 3,1x b. 1

2x

4) a. 4,1x b. 3x

5) a. 15

15,16

x b. 8

8,9

x

6) a. 1

27x b. 9x

7) a. 3

1, 211

x b. 1

2, 22

x

8) a. 4x e b. 0x

9) a. ex e b. 1x

10) a. 4x b. 2x

11) a. No solution b. 1

2x

12) a. 1,0x b. 4x

13) a. 9

16x b. 2x

14) a. 9x b. 2, 6x

15) a. 1x b. No solution

College Algebra 2





log log2 log36 log2x x a. 3 1log log log3 log8 log6x

x b.


5 5 52log 2 log log 3 x x x a. 1 1 1

4 4 4

log 2log 3 log 16 x x b.


2 2 2

12log log 1 log 4

1x

x

a. 21

log 2 log3 log2

x x b.


2 1ln 2 ln 2lnx e

x a. ln 4 ln ln 2 ln ln 1

4

xx x b.


2 4 8 2log 2log 3log log 8x x x a. 2

5 25 1255

1log log 2log 1.5log

2x x x x b.

Final Answers:

1) a. 3x b. 2x

2) a. 4x b. 1x

3) a. 1x b. 1x

4) a. 2

ex b. 1x

5) a. 2x b. 1x

College Algebra 2





2

2 log 3log 5 0x x a. 2 34log log 10log10 0x x b.


8

2log 2log

xx

a. 23

2 31

log3log xx b.


1

2 43

ln 1 2 lnx x

a.

2

2 4

1 32log log

4x

x b.


3log 6 log8 3log 4 x x a. 2 4log 2 log 4 2log 8x x x b.

Final Answers:

1) a. 2.510,10x b. 17.87,0.01x

2) a. 100,0.0001x b. 10,0.001x

3) a. 2 1,x e e b. 8,2x

4) a. 8x b. 2

3

1,2

2

x

College Algebra 2





4log4

xx a. 2log

16x

x b.


1 log 1

100

xx a. 32 2log(3 ) 9

xx b.


41

log2 1

64xx

x a.

2log 41000000xx x b.


103 log

3

1000.1 x

x a.

52 log5 1

625

x

x b.


2

4log10 10000xx x a. 3

37 log

1 log9

xx

x

b.


3

1 log

2 log 110

x

xxx

a. 3log (9 )

33x

x x

b.


log 3

26

110

x

xx

a. log 4

2 84 x

x x b.


5log 6625 4x

x a.

log2

0.011000

x

x

b.

College Algebra 2



Final Answers:

1) a. 1

4,4

x b. 1

16,16

x

2) a. 1

10,100

x b. 1

1,9

x

3) a. 1

8,4

x b. 1

1000,10

4) a. 10x b. 1

625,5

x

5) a. 1

10,10

x b. 1

3,81

x

6) a. 10, 10x b. 9,3x

7) a. 1

10,10 10

x b. 1

2,2

x

8) a. 1

2x b.

1100,

10x

College Algebra 2



Applications - Decay and Growth - Version 1 Questions:

1) A woman deposited $500 in a savings account at an interest rate of 3% compounded annually. Determine how much money will be in the account after:

a. 5 years. b. 2 years. c. 4 months (Assume all months are exactly 1/12 of a year). d. 6 months.

2) Assume that the population of the earth is growing exponentially at a (constant) rate

of 2% per year and that in 1980 it was 4 billion (4,000,000,000). a. What will the population of the earth be in 2020?) b. What was the population of the earth in 1974? c. When will a population of 50 billion be reached?

3) The population in a certain city grows exponentially. In a certain year there were 400

thousand residents and 4 years later there were 440 thousand. a. Find the annual growth rate (as a %). b. After how many years (from that certain year) were there 550thousand residents?

4) A man deposited money in the bank at an interest rate of 4% compounded annually.

After 5 years he had accumulated $5000. a. How much did he initially deposit? b. After how many years will he have accumulated $7000?

5) The number of wild animals at a nature reserve grows exponentially. There were 1000

animals at the initial count. At a second count, 20 months later, there were 1400 wild animals. How many months after the initial count will the reserve have 2000 animals?

6) The radioactive isotope carbon-14 decays exponentially with a half-life of 5750 years a. How many grams of this isotope will remain after 1000 years, if there were 100

grams initially? b. After how many years will there remain just 10 grams of the initial 100 grams?

College Algebra 2



7) In a certain pool there are 240 tons of fish, and the quantity of fish in it increases by 4% each week. In a second pool there are 200 tons of fish, and the quantity of fish in it increases by 10% each week.

a. After how many weeks will both pools have the same quantity of fish? b. After how many weeks will the second pool have twice the quantity of fish as the

first pool?

8) John purchased a car. Assume its value depreciates exponentially. After 4 years its value was $5,000. After an additional 2 years its value dropped to $4,000. What wasthe initial purchase price?

9) The value of a share (of stock) is increasing exponentially and doubles itself in 4 years. How long does it take for the share to triple in value?

10) Assume the value of a car depreciates exponentially. After 4 years it lost 25% of its value. How many years does it take for the car to lose 50% of its value?

Final Answers:

1) a. $579.64 b. $530.45 c. $504.65 d. $507.44

2) a. 8,832,158,654 b. 3,551,885,528 c. 2107 or 2108

3) a. 2.41% b. 13.36 years

4) a. $4,109.63 b. 13.5789 years

5) 41.2 months

6) a. 88.64 grams b. 19,101 years

7) a. 3.25 weeks b. 15.61 weeks

8) $7,812

9) 6.34 years

10) 9.64 years

College Algebra 2



Applications - Decay and Growth - Version 2 Questions:

1) $10,000 are deposited in an account that earns interest at an annual rate of 4%. Determine how much money will be in the account after 26 months if the interest is compounded

Quarterly. a. Monthly. b. Continuously. c.

2) A bank account is opened with an initial deposit of $4000. The annual interest rate is 2%.

How long will it take for the money to double if the interest is compounded

Quarterly. a. Monthly. b. Continuously. c.

3) We deposit $4500 into an account that earns interest at an annual rate of 8%.

How long do weto wait till there are $7000 in the account if the interest is compounded

Continuously? a. 6 times a year? b.

4) The growth of a colony of bacteria is given by the equation, 0

0.215t

tQ Q e where t is in

hours. Given that there are initially 400 bacteria present, a. How many bacteria will there be after two days? b. How long will it take till there are 20,000 bacteria in the colony?

5) A population of bacteria initially grows exponentially. Initially, there are 400 bacteria

present and in 4 days there will be 1,200. a. Determine the growth equation for this population (time is measured in days). b. How long will it take for the population to reach 5,000?

6) The number of wild animals at a nature reserve grows exponentially. There were 1000

animals at the initial count. At a second count, 20 months later, there were 1400 wild animals. How many months after the initial count will the reserve have 2000 animals?

7) The radioactive isotope carbon-14 decays exponentially with a half-life of 5750 years a. How many grams of this isotope will remain after 1000 years, if there were 100

grams initially? b. After how many years will there remain just 10 grams of the initial 100 grams?

College Algebra 2



8) We initially have 200 grams of a radioactive element and in 950 years 90 gramswill remain.

a. Determine the exponential decay equation for this element. b. How long will it take for half of the element to decay? c. How long will it take until there is only 1 gram of the element left?

Final Answers:

1) a. $10,900.63 b. $10,903.76 c. $10,905.33

2) a. 34.74 years b. 34.68 years c. 34.65 years

3) a. 5.52 years b. 5.56 years

4) a. 12,133,303 b. 18.195 hours

5) a. 4400 3t

tQ b. 9.196 days

6) 41.2 months

7) a. 88.64 grams b. 19,101 years

8) a. ln0.45

950200t

tQ e b. 824.65 years c. 6,303.51 years

Logarithms, Logarithmic Equations, Applications - proprep Logarithmic... · 3 3 log 12 10 2 2 x x...

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