Ch. 4 Exponential and Logarithmic Functionsstaff.uob.edu.bh/files/671125478_files/ch04.pdf · Ch. 4 Exponential and Logarithmic Functions 4.1 Exponential Functions SHORT ANSWER. Write

Ch. 4 Exponential and Logarithmic Functions

4.1 Exponential Functions

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

1) The population of a city is given by P = 100,000(1.03)t where t is the number of years after 1988. Find thepopulation in(a) 1988(b) 1989(c) 1990.

2) The population of a city is given by P = 10,000(1.04)t where t is the number of years after 1988. Find thepopulation in(a) 1988(b) 1989(c) 1990.

3) For the function f(x) = 5x,(a) what is the domain of f?(b) what is the range of f?

4)

For the function f(x) = 13

x,

(a) what is the domain of f?(b) What is the range of f?

5) Graph y = f(x) = 3x.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

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6) Graph y = f(x) = (0.5)x.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

7) Suppose that the number of patients admitted into a hospital emergency room during a certain hour of the dayhas a Poisson distribution with mean 3. Find the probability that during that hour there will be exactly twoemergency patients. Assume that e-3 = 0.05.

8) If $2000 is invested for 3 years at 8% compounded quarterly, find(a) the compound amount and(b) the compound interest.

9) If $400 is invested for 2 years at 6% compounded semiannually, find(a) the compound amount and(b) the compound interest.

10) Suppose $5000 is deposited in a savings account that earns 10% compounded semiannually. What is the valueof the account at the end of 6 years? Assume no other deposits or withdrawals.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

11) The population of a city is given by P = 1,000,000(1.02)t where t is the number of years after 1987. Thepopulation in 1989 was

A) 1,002,000. B) 1,020,000. C) 1,004,000. D) 1,040,000. E) 1,040,400.

12) Of the following the best approximation of e is

A) 1.4. B) 1.8. C) 2.3. D) 2.7. E) 3.1.

13) Which of the following is true? If f(x) = ex, then

A) both the domain and range of f are all real numbers.

B) both the domain and range of f are all positive real numbers.

C) the domain of f is all positive real numbers and the range is all real numbers.

D) the domain of f is all real numbers and the range is all positive real numbers.

E) the domain of f is all real numbers except zero and the range is all real numbers.

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14)

The above graph is best represented by

A) y = 4x B) y = 14

xC) y = e4 D) y = log4 E) y = ln 4

15) If $1000 is invested for 2 years at 6% compounded quarterly, then the compound amount at the end of theperiod is

A) $1120.00. B) $1123.60. C) $1126.49. D) $1141.23. E) $1593.84.

16) If $500 is invested for 3 years at 7% compounded semiannually, then the compound interest at the end of theperiod is

A) $112.52. B) $114.63. C) $120.37. D) $122.56. E) $150.36.

17) If $10,000 is invested at 16% compounded quarterly, then the compound amount at the end of six years is

A) $21,173.75. B) $24,278.09. C) $25,633.04. D) $26,678.42. E) $26,987.33.


18) Find the equations of the graph that is obtained from the graph of y = ex shifted(a) 3 units down;(b) 3 units to the right.

19) A radioactive element is such that N grams remain after t hours, where N = 20e-0.028t. How many gramsremain after 30 hours?

20) A trust fund is being set up by a single payment so that at the end of 30 years there will be $20,000 in the fund.If the interest rate is 8% compounded quarterly, how much money should be paid initially into the trust fund?

21) A trust fund is being set up by a single payment so that at the end of 5 years there will be $10,000 in the fund. If

the interest rate is 3 34% compounded quarterly, how much money should be paid initially into the trust fund?

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22) By looking at the graph of y = ex, sketch a graph of y = ex+2 - 3.

23) The number of bacteria in a culture is doubling every hour. Currently the culture has 128 bacteria. Make a tableof values for the number of bacteria present each hour for 0 to 4 hours. For each hour write an expression forthe number of bacteria as a product of 128 and a power of 2. Use the expressions to make an entry in your tablefor the number of bacteria after t hours. Write a function N for the number of bacteria after t hours.

24) The number of bacteria in a culture is growing by 40% every hour. Currently the culture has 500 bacteria. Makea table of values for the number of bacteria present each hour for 0 to 4 hours. For each hour write anexpression for the number of bacteria as a product of 500 and a power of 1.4. Use the expressions to make anentry in your table for the number of bacteria after t hours. Write a function N for the number of bacteria after thours.

25) A certain medicine reduces the bacteria present by 25% each day. Currently 28,000 bacteria are present. Make atable of values for the number of bacteria present each day for 0 to 4 days. For each day write an expression forthe number of bacteria as a product of 28,000 and a power of 0.75. Use the expressions to make an entry in yourtable for the number of bacteria after t days. write a function N for the number of bacteria after t days.

26) A graphical look at Bacteria Growth: If 100 bacteria are present at the start, the number of bacteria in a culturewhich changes by constant factor f every hour is given by N(t) = 100(ft). Use a graphing calculator to graph thisfunction for various values of f where f > 0. Describe how the graphs where 0 < f < 1 differ from the graphswhere f > 1. How does the number of bacteria change when 0 < f < 1? How does the number of bacteria changewhen f > 1? Describe the graph where f = 1. How does the number of bacteria change when f = 1?

27) Assume an investment is guaranteed to triple every decade.(a) Make a table of the factor of increase in the investment at each decade for 0 to 3 decades. For each decade,write an expression for the factor of increase as a power of some base.(b) What base did you use? How does that base relate to the problem?(c) Use your table to graph the factor of increase as a function of decades.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(d) Use your graph to estimate when the investment will have grown by a factor of 15.

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28) Assume the amount of paper being recycled quadruples every year.(a) Make a table of the factor of increase in the amount of paper being recycled at each year for 0 to 3 years.For each year, write an expression for the factor of increase as a power of some base.(b) What base did you use? How does that base relate to the problem?(c) Use your table to graph the factor of increase as a function of years.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(d) Use your graph to estimate when the recycling will have grown by a factor of 50.

29) An investment increases by 10% every year. Write a function for the factor of increase in the investment as afunction of years. Use a graphing calculator to graph your function. Use the graph to estimate when theinvestment will double.

30) The amount of plastic being recycled increases by 30% every year. Write a function for the factor of increase inplastic recycling as a function of years. Use a graphing calculator to graph your function. Use the graph toestimate when the amount of recycling will triple.

31) Assume the value of a sailboat depreciates by 18 every year.

(a) Make a table of the factor of decrease in the value of the sailboat for 0 to 3 years. For each year, write anexpression for the factor of decrease in the value of the sailboat as a power of some base.(b) What base did you use? How does the base relate to the problem?(c) Use your table to graph the factor of decrease as a function of years.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(d) Use your table to guess when your boat will be worth 25% as much as its original price.

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32) Assume the value of a computer depreciates by 30% every year.(a) Make a table of the factor of decrease in the value of the computer for 0 to 3 years. For each year, write anexpression for the factor of decrease in the value of the computer as a power of some base.(b) What base did you use? How does the base relate to the problem?(c) Use your table to graph the factor of decrease as a function of years.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(d) Use your graph to guess when your computer will be worth a tenth of its original price.(e) Using the table you made, write a function for the depreciation as a function of years. Use a graphing calculator to graph your function. Use the graph to estimate when the computer will be worth a tenth of its original value. Compare this answer with the guess you made from your graph.

33) Assume the value of an RV depreciates by 22% every year.(a) Make a table of the factor of decrease in the value of the RV for 0 to 3 years. For each year, write anexpression for the factor of decrease in the value of the RV as a power of some base.(b) What base did you use? How does the base relate to the problem?(c) Use your table to graph the factor of decrease as a function of years.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(d) Use your graph to guess when your RV will be worth a tenth of its original price.(e) Using the table you made, write a function for the depreciation as a function of years. Use a graphing calculator to graph your function. Use the graph to estimate when the RV will be worth a tenth of its original value. Compare this answer with the guess you made from your graph.

34) Assume your $2000 investment,which is guaranteed to triple every decade, has a one time $10 service charge.(a) Make a table of the value of your investment without the service charge at each decade from 0 to 3.(b) Make a table of the value of your investment with the service charge deducted at each decade from 0 to 3.(c) How could you use a graph of (a) to make a graph of (b)?

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35) Assume your savings consist of a $5000 investment which is guaranteed to increase by 7% every year and $800cash in a safe at your home.(a) Make a table of the value of your investment at 0 to 3 years.(b) Make a table of the value of total savings at 0 to 3 years.(c) How could you use the graph of (a) to make a graph of (b)? Verify your answer using a graphingcalculator.

36) Sean and Carley have bacterial infections. Sean was given medicine which reduces the number of bacteriahourly by 20%. 5 hours later Carley began the same treatment.(a) If y = 0.8t represents the multiplicative decrease of bacteria for Carley, write an equation using the samereference that represents the multiplicative decrease in the bacteria in Sean.(b) If a doctor had a graph of the multiplicative decrease of bacteria for Carley, how could she use it to graphthe multiplicative decrease of the bacteria in Sean? Verify your answer using a graphics calculator.

37) Suppose $2000 is invested at 6.5% compounded annually.(a) Find the value of the investment after 5 years.(b) Find the value of the interest which was earned over the first 5 years.

38) Suppose $2000 is invested at 6.5% compounded annually.(a) Find the value of the investment after 10 years.(b) Find the value of the interest which was earned over the first 10 years.

39) Suppose $20,000 is invested at 6.5% compounded annually.(a) Find the value of the investment after 5 years.(b) Find the value of the interest which was earned over the first 5 years.

40) Suppose $2000 is invested at 6.5% compounded quarterly.(a) Find the value of the investment after 5 years.(b) Find the value of the interest which was earned over the first 5 years.

41) Suppose $2000 is invested at 6.5% compounded monthly.(a) Find the value of the investment after 5 years.(b) Find the value of the interest which was earned over the first 5 years.

42) Suppose $2000 is invested at 6.5% compounded daily (exclude extra day for leap year).(a) Find the value of the investment after 5 years.(b) Find the value of the interest which was earned over the first 5 years.

43) A company is downsizing and expects the number of employees to shrink at the rate of 3% per month.Currently the company employs 30,000 people.(a) How many people are expected to be employed with this company in 1 year?(b) Use a graphing calculator to predict the number of months until the number of employees will be half itscurrent size.

44) A town of 1400 is growing at the rate of 8% per year. If this rate continues, what will the population of thistown be in 20 years?

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45) The number of yearly visitors to a resort has been shrinking at the rate of 6%. Currently the resort gets 120,000tourists each year.(a) If this rate continues, how many tourists will they get in 15 years?(b) Use a graphing calculator to predict the number of years until the number of tourists will be less than20,000.

46) The multiplicative decrease in purchasing power P after t years of inflation at 3% can be modeled by P =

e-0.03t. Graph the decrease in purchasing power as a function of t years.

x

y

x

y

47) Graph the population for a city with 80,000 people as a function of years if the growth is modeled byP = 80,000e0.04t.

x

y

x

y

48) Graph f(x) = 2x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

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49) Graph g(x) = 13x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

50) Graph h(x) = -2x+3 - 1

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

51) Graph f(x) = 3 23x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

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52) Graph f(x) = ex - 2

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

4.2 Logarithmic Functions



1) Express log2 18 = -3 in exponential form.

2) Express log4 64 = 3 in exponential form.

3) Express 34 = 81 in logarithmic form.

4) Express 10-3 = 11000

in logarithmic form.

5) Graph y = f(x) = log4 x.

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

6) For the function f(x) = log4 x,(a) what is the domain of f?(b) What is the range of f?

7) Evaluate and simplify: log 100

8) Evaluate and simplify: ln e3

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9) Evaluate and simplify: log (0.1)

10) Evaluate and simplify: log71

11) Evaluate and simplify: ln e

12) Evaluate and simplify: log3 181

13) Evaluate and simplify: log2(-2)

14) Evaluate and simplify: log6 6

15) Find x: log3 x = 3

16) Find x: logx = 0

17) Find x: ln x = 1

18) Find x: log4 x = 2

19) Find x: log2 x = -3

20) Find x: log x = 3

21) Find x: ln x = -2

22) Find x: log x = -2

23) Find x: log6 36 = x

24) Find x: log5 125 = x

25) Find x: log 100,000 = x

26) Find x: log4 2 = x

27) Find x: log 0.01 = x

28) Find x: ln e3 = x

29) Find x: logx16 = 4

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30) Find x: log2(x + 4) = 3

31) Find x: logx(4x - 1) = 1

32) Find x: logx(4x -3) = 2

33) Find x: log4 46 = x

34) Find x and express your answer in terms of natural logarithms: e4x = 2

35) Find x and express your answer in terms of natural logarithms: 2e3x = 6

36) A radioactive substance decays according to the equation N = 10e-0.04t, where N is the number of milligramspresent after t days. Find the half-life of the substance.


37)

The above graph is best represented by

A) y = 4x B) y = 14

xC) y = e4 D) log4 x E) y = ln 4

38) Which of the following is true? If f(x) = ln x, then

A) both the domain and the range of f are all real numbers.

B) both the domain and the range of f are all positive real numbers.

C) the domain of f is all positive real numbers and the range is all real numbers.

D) the domain of f is all real numbers and the range is all positive real numbers.

E) the domain of f is all real numbers except zero and the range is all real numbers.

39) log5 125 =

A) 0 B) 1 C) 2 D) 3 E) 4

40) log 0.001 =

A) -1 B) -2 C) -3 D) -4 E) -5

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41) The value of log3 0 is

A) 0 B) 1 C) 3 D) -1 E) not defined

42) log5 125 =

A) -5 B) -2 C) -4 D) 12

E) - 12

43) If log4 x = -3, then x =

A) 164

B) 64 C) -12 D) 81 E) 181

44) If log2(4x + 1) = 3, then x =

A) 34

B) 54

C) 74

D) 94

E) 114

45) If log2(x + 3) = -2, then x =

A) -1 B) -7 C) 34

D) - 43

E) - 114

46) If 2e3x - 5 = 3, then x =

A) 43

B) ln 43

C) 3ln 4

D) ln 34

E) 13ln 7

2

47) If a radioactive substance decays according to the equation N = 40e-0.03t, where N is the number of milligramspresent after t days, then the half-life, in days, of the substance is given by

A) - ln 20.03

. B) ln 20.03

. C) 0.03ln 2

. D) - 0.03ln 2

. E) ln 240

.


48) Find x: logx(6 - 4x - x2) = 2

49) A manufacturerʹs supply equation is p = log 2 + 3q5 where q is the number of units supplied at price p per unit.

At what price will the manufacturer supply 200 units?

50) How long will it take for $100 to amount to $200 at an interest rate of 10% compounded annually? Give youranswer to 2 decimal places.

51) Solve for x: logx 3 = 12

52) Solve for x: logx(2x + 3) = 2

53) Solve for x: loge2 x = 5

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54) Find the domain of y = log3(x + 1)

55) True or False: If loga x = logb x, then a = b.

56) Solve the equation for x in terms of y: 1.2y = log x1.2 × 105

57) Solve for x: log(98 - x + x2) = 2

58) Solve for x: log(x2 + 4x + 104) = 2

59) Solve for x: logx y = 3

60) The work done by 1 kilogram sample of nitrogen as its volume changes from an initial value Vi to a final value

of Vf during a constant temperature process is given by W = 3.2×102lnVfVi

. If such a sample expands from a

volume of 3 liters to a volume of 5 liters, determine the work done by the gas.

61) If money is invested at 7% compounded annually and the current amount is 3 times the amount first invested,then the situation can be represented by 3 = (1.07)t. Represent this equation in logarithmic form. What does trepresent?

62) If the yeast has been decreasing by 80% hourly and the current amount is 178,125

of the amount first measured,

then the situation can be represented by 178,125

= 15

t. Represent this equation in logarithmic form. What

does t represent?

63) If a car is depreciating by 12.5% each year and the current amount is 0.5 its original value, then the situation

can be represented by 0.5 = 78

t. Represent this equation in logarithmic form.

64) An earthquake measuring 5.8 on the Richter scale can be represented by 5.8 = log II0 where I is the intensity of

the earthquake and I0 is the intensity of a zero-level earthquake. Represent this equation in exponential form.

65) If the pH of a substance is 9.2, then the concentration of hydrogen ions h in gram-atoms per liter can be

represented by 9.2 = log 1h. Represent this equation in exponential form.

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66) R = log II0 gives the Richter Scale measurement of an earthquake with intensity I where I0 is the intensity of a

zero-level earthquake. Make a graph which converts the intensity of the earthquake as compared to azero-level earthquake to the Richter Scale measurement.

67) Suppose an investment triples every decade. Graph the number of decades invested as a function of themultiplicative increase in original investment. Label the graph with the name of the function.

68) Suppose an investment increases by 10% every year. . Graph the number of years invested as a function of themultiplicative increase in original investment. Label the graph with the name of the function.

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69) Suppose a computer decreases in value by 50% every year. Graph the number of years it is owned as a functionof the multiplicative decrease in its original value. Label the graph with the name of the function.

70) Suppose a garbage company has found that garbage per family has decreased by 10% every year since the firstyear they started a curb-side recycling program. Graph each year as a function of the multiplicative decrease ingarbage since the first year of the curb-side recycling program. Label the graph with the name of the function.

71) The magnitude (Richter Scale) of an earthquake is given by R = log II0 where I is the intensity of the

earthquake and I0 is the intensity of a zero-level reference earthquake. II0 represents how many times greater

the earthquake is than the reference earthquake. Find the magnitude of an earthquake that is 200 times theintensity of a zero-level earthquake.

72) The number of years it takes for an amount which is invested at an annual rate of p and compounded

continuously to become m times as large is given by t = ln(m)p

. How long does it take an investment to

quadruple if it is invested at an annual rate of 7% and compounded continuously?



the earthquake is than the reference earthquake. Find the magnitude of an earthquake that is 2,000,000 timesthe intensity of a zero-level earthquake.

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74) The number of years it takes for an amount which is invested at an annual rate of p and compounded

continuously to become m times as large is given by t = ln(m)p

. How long does it take an investment to triple if

it is invested at an annual rate of 8% and compounded continuously?

75) The number of years it takes for an amount which is invested at an annual rate of 7% and compounded

continuously to become m times as large is a function of the enlargement factor given t(m) = ln(m)0.07

. Use a

graphics calculator to find how many times larger (to the nearest whole number increment) the investment willbe in 10, 20, 30, and 40 years.



than the earthquake is than the reference earthquake. If an earthquake measured 7.5 on the Richter Scale, howmany more times intense is it than a barely-felt earthquake?



than the earthquake is than the reference earthquake. If an earthquake measured 4.2 on the Richter Scale, howmany more times intense is it than a zero-level earthquake?

78) The multiplicative increase m of an investment which is invested at an annual rate of p and compoundedcontinuously for a time t is given by m= ept. If your annual rate is 6.75%, how many years will it take to doubleyour investment?

79) Graph f(x) = log2x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

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80) Consider g(x) = -log3(x + 2)(a) Graph g(x)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(b) Find the domain of g(x)

81) Consider h(x) = 2ln(x) - 3(a) Graph h(x)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(b) Find the domain of h(x)

4.3 Properties of Logarithms



1) Evaluate and simplify: 10log 4

2)

Evaluate and simplify: log5352

3) Evaluate and simplify: log2 1 + log2 2

4) Evaluate and simplify: ln e2 + ln 1

5) Assume that log 6 = 0.7782. Determine the value of log 36.


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8) Assume that log 6 = 0.7782. Determine the value of log(0.6).



11) Assume that log 4 = 0.6021. Determine the value of log 116

.

12) Assume that log 3 = 0.4771. Determine the value of log (0.09).

13) Assume that log 5 = 0.6690 and log 6 = 0.7782. Determine the value of log 30.





18) Write the following in terms of ln x and ln(x + 1): ln x4

(x + 1)3

19) Write the following in terms of ln(x + 2) and ln(x + 4): ln (x + 2)2(x + 4)

20) Write the following in terms of ln(x + 2) and ln(x + 4): ln (x + 2) x + 4

21)

Write the following in terms of ln(x + 2) and ln(x + 4): ln4(x + 2)(x + 4)3

22) Write the following in terms of ln(x + 2) and ln(x + 4): ln x + 2x + 4

23) Write the following in terms of ln(x + 2) and ln(x + 4): ln 1(x + 2)2(x + 4)3

24) Write the following in terms of ln x, ln(x - 3), and ln(x + 1): ln x + 1x2(x - 3)

25) Write the following in terms of ln x, ln(x - 3), and ln(x + 1): ln x3(x - 3)(x + 1)2

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26) Write the following in terms of ln x, ln(x - 3), and ln(x + 1): ln x(x - 3)2

x + 1

27) Express 2 ln 3 - ln 4 as a single logarithm.

28) Express ln 4 - 12(ln 2 + ln 3) as a single logarithm.

29) Express 2(ln 4 + ln 3 - 3 ln 2) as a single logarithm.

30) Express 1 + ln x as a single logarithm.

31) Express 2 log(x) - 3 log(x + 7) as a single logarithm.

32) Solve for x: ln e2x = x

33) Solve for x: eln(3x+4) = 10

34) Solve for x: 3log3x+log34 = 8

35) Solve for x: eln(4x) = 20

36) Solve for x: e2ln(2x) = 4

37) If ln 2 = 0.7 and ln 5 = 1.6, find log2 5.

38) If log 4 = 0.6 and log 7 = 0.8, find log4 7.

39) Write log6 x in terms of natural logarithms.

40) Write log(x + 3) in terms of natural logarithms.

41) Write log2(x + 4) in terms of common logarithms.


42)

log6564 =

A) 54

B) - 54

C) 45

D) - 45

E) 0

43) log 10,000 - 6 log 10 =

A) 10,0006 10

B) -6log 10 C) 0 D) 4 E) 1

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44) log300100

=

A) 3

B) log 3

C) log 200

D) 2 + log32

E) none of the above

45) ln e - ln 1 =

A) 0 B) 1 C) 6 D) n(e - 1) E) ee-1

46) ln 6 + ln 16 + ln e2 =

A) 0 B) 1 C) 2 D) 3 E) 4

47) If log 3 = 0.4771, then log 0.3 =

A) 0.5229. B) -0.5229. C) 1.4441. D) -1.4771. E) -4.771.

48) If log 2 = 0.3010 and log 3 = 0.4771, then log 18 =

A) 0.0685. B) 0.1761. C) 1.2552. D) 1.9030. E) 2.3343.


A) 0.2136. B) 0.4259. C) 1.3801. D) 1.8927. E) 2.8239.


A) 0.1761. B) 1.1303. C) 1.7323. D) 3.1761. E) 04.283.

51) If log(x + 3)4 = 4, then x can equal

A) -3. B) 7. C) 12. D)44. E) 1

10.

52) If ln e5x-4 = 6, then x =

A) 45. B) ln 4

5. C) e4/5. D) 1. E) 2.

53) If 10logx = 5, then x =

A) 0. B) 1. C) 5. D) 52. E) log 5.

54) If e ln(4x+3) = 7, then x =

A) 0. B) 1. C) 2. D) -2. E) 14 ln 7

3.

Page 177

55) ln xyz =

A) ln x - 12(ln y + ln z)

B) ln x - 12(ln y - ln z)

C) ln x + 12(ln y - ln z)

D) ln x - ln y + ln z

E) ln xln y ln z

56) ln (xy)2

z3 =

A) ln(xy)2

ln z3

B) 2 ln x ln y - 3 ln z

C) ln 2 + ln x + ln y - ln 3 - ln z

D) 2(ln x + ln y) - 3 ln z

E) ln 2 ln x ln y - ln 3 ln z

57) Writing 2 ln x - 13ln y + 4 ln z as a single logarithm gives

A) ln 2x - 13y + 4z .

B) ln(x2 - 3y + z4).

C) ln x2z4

y3.

D) ln x2z43y.

E) ln(x2 + z4)

ln3y

.

58) Writing 16 [ln x - 2(ln y + 2 ln z)] as a single logarithm gives

A) ln6x

yz2. B) ln

6x

y2z4. C) ln 6 xz

4

y2. D) ln 6 x

y2z2. E) ln 6 x

y2z4.

Page 178

59) If log 2 = 0.3010 and log 3 = 0.4771, then log2 3 =

A) 0.4471 - 0.3010

B) 0.3010 - 0.4771

C) 0.47710.3010

D) 0.30100.4771

E) (0.3010)(0.4771)

60) Changing log7(4x) to natural logarithms gives

A) ln(4x)ln 7

. B) ln 7ln(4x)

. C) ln (4x). D) ln(4x)ln e

. E) ln eln(4x)

.


61) Assume log3 x = 2; log3 y = .12. Find log33xy2.

62) Assume ln x = 2; ln y = 7. Find ln(xy2).

63) Given ln x = 7.1; ln y = 8.2, find ln(x3y7)

64) Given ln x = 7.1; ln y = 8.2, find ln x3

y7.

65) Solve for x: aloga x + loga4 = 8

66) Simplify: 105log x

67) Simplify: e2 ln x - 3 ln y

68) The population of India was 651 million in 1980 and has been growing at a rate of 2% per year. The populationt years later is approximated by N(t) = 651e.02t. Estimate the population in India in the year 2010.

69) The population in the state of California is 24 million today. It is increasing at a rate of 0.9% per year. Thepopulation t years from now is given by the rule P = 24e.009t. After how many years will the populationdouble?

70) Use a graphing calculator to approximate the solution 2x - 3x = 20.

71) How much more on the Richter Scale is an earthquake with intensity 300,000 times the intensity of a zero-levelearthquake than an earthquake with intensity 150,000 times the intensity of a zero-level earthquake? Write asan expression involving logarithms. Simplify by combining logarithms and then use a calculator to evaluate.

Page 179

72) What is the sum of the Richter Scale measurement of an earthquake which is 37,000 times the intensity of azero-level earthquake and an earthquake with intensity 1000 times the intensity of a zero-level earthquake?Write as an expression involving logarithms. Simplify by combining logarithms and then use a calculator toevaluate.

73) What is the sum of the Richter Scale measurement of an earthquake which is 250,000 times the intensity of azero-level earthquake and an earthquake with intensity twice the intensity of a zero-level earthquake? Writeas an expression involving logarithms. Simplify by combining logarithms and then use a calculator to evaluate.

74) If an earthquake is 10x+2 times as intense as a zero-level earthquake, what is its measurement on the RichterScale? Write as logarithmic expression and simplify.

75) If an earthquake is 103(x-1) times as intense as a zero-level earthquake, what is its measurement on the RichterScale? Write as logarithmic expression and simplify.

76) If an earthquake is 2x · 5x times as intense as a zero-level earthquake, what is its measurement on the RichterScale? Write as a logarithm expression and simplify.

77) What power of 3 is 36?





82) Steve wants to use his graphing calculator to check his sketch of y = log5 x but his calculator does not makelog5 calculations. Find two equations he could use.

83) April wants to use her graphing calculator to check her sketch of y = log8 x but her calculator does not makelog8 calculations. Find two equations she could use.

84) Brett wants to use his graphing calculator to check his sketch of y = log0.5 x but his calculator does not makelog0.5 calculations. Find two equations he could use. Use a graphing calculator to confirm that the twoequations are equivalent.

85) Hannah wants to use her graphing calculator to check this sketch of y = log12 x but her calculator does notmake log12 calculations. Find two equations she could use. Use a graphing calculator to confirm that the twoequations are equivalent.

Page 180

86) Use the change base formula and your graphing calculator to graphf(x) = log3 x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

87) Use the change base formula and your graphing calculator to graphf(x) = -3log2 (x + 4) + 1

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

4.4 Logarithmic and Exponential Equations



1) Find x and express your answer in terms of common logarithms: 4x = 3

2) Find x and express your answer in terms of common logarithms: 102x-3 = 4

3) Find x and express your answer in terms of natural logarithms: 2-x - 3 = 8

4) Solve for x: ln(x + 3) = ln(2x)

5) Solve for x: log x = log 3 + 2 log 4

6) Solve for x: 42x = 2

7) Solve for x: ln x + ln 3 = ln(x + 1)

8) Solve for x: ln(x + 1) - ln x = ln 2

Page 181

9) Solve for x: log(x + 1) - log(x - 2) = 1

10) Solve for x: 4x+1 = 83x

11) Solve for x: log2(x - 4) + log2 3 = log2 x

12) If p = 320-q, by using common logarithms express q in terms of p.

13) Solve for t: 300 = 500(1 - e0.2t). Assume that ln(0.4) = -0.9.

14) The demand equation for a product is given by q = 1000 - 3p. Solve for p and express your answer in terms ofcommon logarithms.


15) If ln x + ln 2 = ln 5, then x =

A) 13. B) 3. C) 2

5. D) 5

2. E) 5.

16) If 22x+1 = 8x-3, then x =

A) 10. B) -4. C) 6. D) log2 3. E) -7.

17) If p = 54-q, then in terms of common logarithms, q =

A) log54-p. B) p - 4 log 5. C) 4 - log plog 5

. D) 5 - log plog 4

. E) 5 - log 5log p

.

18) Initially, three are 50 milligrams of a radioactive substance. The substance decays according to the equation N =

50e-0.01t, where N is the number of milligrams present after t hours. The number of hours it takes for 10milligrams to remain is

A) -

15

ln(0.01). B) -

ln 15

0.01 . C) - ln(0.01)

5. D)

15

ln(-0.01). E) 50e-0.1.

19) If log4(x + 6) = 2 - log4 x, then x =

A) -4. B) 2. C) 4. D) 8. E) 10.


20) Solve for x: log3(x + 7) - 3log3 2 = log3 x

21) Solve for x: (2 + x)5 = 129.3

22) Solve for x: 2log2x + log25 = 7

23) Solve for x: 22x - 2 · 2x - 3 = 0

Page 182

24) Find x if 7e3x = 1

25) Solve: 10log x3 = 27

26) Your friend started a savings plan with 2 pennies and each day he doubled the amount saved. Later youstarted a savings plan with 4 pennies and each day you quadrupled the amount saved. Your friend has beensaving for 6 less than 3 times as many days as you. In two days you will put the same amount in savings asyour friend. How many days have you been saving?

27) April took a number and multiplied it by a power of 9. Bob started with the same number and got the sameresult when he multiplied it by 27 raised to a number which was four less than the exponent that April used.What power of 9 did April use?

28) Your friend started a savings plan with 3 pennies and each day she saved three times the amount she saved theprevious day. Later you started a savings plan with 9 pennies and each day you saved 9 times the amount yousaved the previous day. On the day that your friend had been saving for 8 less than 4 times as many days asyou, you saved the same amount as your friend. How many days have you been saving?

29) The value of an investment of $1000 earning 8% compounded yearly is given by A = 1000(1.08)t, where t is thenumber of years it has been invested .If the amount of your investment is now $4000, how long has it beeninvested?

30) The sales manager of a department store finds that its daily sales begins to fall after the end of a promotionalcampaign. The sales in dollars as a function of the number of days after the campaignʹs end is given by S(d) =

36,000 98 -0.1d

. If she does not want sales to drop below 30,000 per day before starting a new campaign, when

should she start a new campaign?

31) The value of an investment of $3000 earning 7.25% compounded yearly is given by A = 3000(1.0725)t, where t isthe number of years it has been invested. If the amount of your investment is now $10,000, how long has itbeen invested?

32) The demand function for a product is p = 45 5-q/10 where q is the number of units and p is the price of oneunit. At what price will the demand be 5 units? How many units will be demanded if the price is $12.42?




36) An earthquake which is 48,000 times as intense as a zero-level earthquake has a magnitude on the Richter Scalewhich is 1.7 less than the intensity of another earthquake. What is the intensity of the other earthquake?

Page 183

37) An earthquake which is 520,000 times as intense as a zero-level earthquake has a magnitude on the RichterScale which is 2.7 more than the intensity of another earthquake. What is the intensity of the other earthquake?

38) An earthquake which is 7200 times as intense as a zero-level earthquake has a magnitude on the Richter Scalewhich is 3.6 less than the intensity of another earthquake. What is the intensity of the other earthquake?

Page 184

Ch. 4 Exponential and Logarithmic FunctionsAnswer Key

4.1 Exponential Functions1) (a) 100,000

(b) 103,000(c) 106,090

2) (a) 10,000(b) 10,400(c) 10, 816

3) (a) all real numbers(b) all positive real numbers

4) (a) all real numbers(b) all positive real numbers

5)

6)

7) 0.2258) (a) $2536.48

(b) $536.48.9) (a) $450.20

(b) $50.2010) $8979.2811) E12) D13) D14) A15) C16) B17) C18) (a) y = ex - 3

(b) y = ex-319) 8.634 gms20) $1857.8521) $8297.5422) Shift the graph 2 units to the left and 3 units down.

Page 185

23)

Hours Bacteria Expression0 128 128 · 20

1 256 128 · 21

2 512 128 · 22

3 1024 128 · 23

4 2048 128 · 24

t 128 · 2t

Equation N(t) = 128 · 2t

24)

Hours Bacteria Expression0 500 500(1.4)0

1 700 500(1.4)1

2 980 500(1.4)2

3 1372 500(1.4)3

4 1920.8 500(1.4)4

t 500(1.4)t

Equation N(t) = 500 · 1.4t

25)

Days Bacteria Expression0 28,000 28,000(0.75)0

1 21,000 28,000(0.75)1

2 15,750 28,000(0.75)2

3 11,812.5 28,000(0.75)3

4 8859.4 28,000(0.75)4

t 28,000(0.75)t

Equation N(t) = 28,000(0.75)t26) The graphs for 0 < f < 1 rapidly decrease to the x-axis. The graphs for f > 1 rapidly increase from the x-axis. When 0 <

f < 1, the number of bacteria is decreasing. When f > 1, the number of bacteria is increasing. The graph for f = 1 is ahorizontal line. The number of bacteria does not change when f = 1.

27) (a)Multiplicative

Decade Increase Expression0 1 30

1 3 31

2 9 32

3 27 33(b) 3; The investment triples every decade.(c)

(d) about 2.5 decades or 25 years

Page 186


Year Increase Expression0 1 40

1 4 41

2 16 42

3 64 43(b) 4; The recycling quadruples every year.(c)

(d) about 2.8 years

29) f(t) = 1.1t; about 7.3 years30) f(t) = 1.3t; about 4.2 years31) (a)

MultiplicativeYear Decrease Expression

0 1 78

0

1 0.875 78

1

2 0.766 78

2

3 0.670 78

3

(b) 78; The boat depreciates by 1

8 every year 1 - 1 · 1

8 = 1 - 1

8 = 7

8.

(c)

(d) Answers will vary.

Page 187


Year Decrease Expression0 1 0.70

1 0.7 0.71

2 0.49 0.72

3 0.343 0.73(b) 0.7; The computer depreciates by 30% every year (1 - 1(0.3) = 1 - 0.3 = 0.7).(c)

(d) Answers will vary.(e) f(t) = 0.7t; about 6.5 years


Year Decrease Expression0 1 0.780

1 0.78 0.781

2 0.6084 0.782

3 0.474552 0.783(b) 0.78; The RV depreciates by 22% every year (1 - 1(0.22) = 1 - 0.22 = 0.78).(c)

(d) Answers will vary.(e) f(t) = 0.78t; about 9.3 years

34) (a), (b)Value of Investment Value of Investment

Decade Without Service Charge With Service Charge0 $ 2,000 $ 1,9901 6,000 5,9902 18,000 17,9903 54,000 53,990

(c) Shift the graph down 10 units.

Page 188

35) (a), (b)Decade Value of Investment Value of Total Savings

0 $5,000.00 $5,800.001 5,350.00 6,150.002 5,724.50 6,524.503 6,125.22 6,925.22

(c) Shift the graph up 800 units.36) (a) y = 0.8t+5

(b) Shift the graph 5 units to the left.37) (a) $2740.17

(b) $740.1738) (a) $3754.27

(b) $1754.2739) (a) $27,401.73

(b) $7,401.7340) (a) $2760.84

(b) $760.8441) (a) $2765.63

(b) $765.6342) (a) $2767.98

(b) $767.9843) (a) 20,815 people

(b) about 23 months44) 6525 people45) (a) 47,435 tourists

(b) about 29 years46)

47)

Page 189

48) Graph of f(x) = 2x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

49) Graph of g(x) = 13x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

50) Graph of h(x) = -2x+3 - 1

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

51) Graph of f(x) = 3 23x

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

Page 190

52) Graph of f(x) = ex - 2

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

4.2 Logarithmic Functions1) 2-3 = 1

8

2) 43 = 643) log3 81 = 4

4) log 11000

= -3

5)

6) (a) all positive real numbers(b) all real numbers

7) 28) 39) -110) 0

11) 12

12) -413) not defined14) 115) 2716) 117) e18) 16

19) 18

20) 100021) e-2

22) 1100

23) 2

Page 191

24) -225) 5

26) 12

27) -228) 329) 230) 4

31) 13

32) 1, 333) 6

34) ln 24

35) ln 33

36) ln 20.04

≈ 0.693150.04

≈ 17.33 days

37) D38) C39) D40) C41) E42) B43) A44) C45) E46) B47) B48) 149) $2.0950) 7.27 years51) x = 952) x = 353) x = e1054) all real numbers greater than -155) False. Example: x = 1, a = 7, b = 3.56) 1.2 × 101.2y+557) x = -1, 258) x = -259) x = y1/360) 163.464261) t = log1.07 3; the number of times the investment has been compounded.

62) t = log1/5 1

78,125; the number of hours the yeast has been decreasing

63) t = log7/8 0.5; the number of years the car has been depreciating

64) II0 = 105.8

65) 1h = 109.2

Page 192

66)

67)

68)

69)

Page 193

70)

71) approximately 2.372) approximately 19.8 years.73) approximately 6.374) approximately 13.7 years.75) approximately 2 times as large; approximately 4 times as large; approximately 8 times as large; approximately 16

times as large76) 107.5 = 31,622,776.6 times as intense77) 104.2 = 15,848.9 times as intense78) approximately 10.3 years79) Graph of f(x) = log2x

x-10 -5 5 10

y10

5

-5

-10

(2, 1)

(4, 2)(8, 3)

(1, 0)

x-10 -5 5 10

y10

5

-5

-10

(2, 1)

(4, 2)(8, 3)

(1, 0)

80) a) Graph of g(x) = -log3(x + 2)

x-10 -5 5 10

y10

5

-5

-10

(-1, 0)

(1, -1)(7, -2)

x-10 -5 5 10

y10

5

-5

-10

(-1, 0)

(1, -1)(7, -2)

b) The domain of g(x) is all real numbers greater than -2.

Page 194

81) a)

x-10 -5 5 10

y10

5

-5

-10

(1, -3)

(4, -.227)(9, 1.394)

x-10 -5 5 10

y10

5

-5

-10

(1, -3)

(4, -.227)(9, 1.394)

b) The domain of h(x) is all positive real numbers.4.3 Properties of Logarithms

1) 4

2) 23

3) 14) 25) 1.55646) 2.66907) 0.22308) -0.22189) 1.341310) 2.602111) -1.204212) -1.045813) 1.447214) 0.109215) 1.079216) 1.556317) 0.125018) 4 ln x - 3 ln (x + 1)19) 2 ln(x + 2) + ln(x + 4)

20) ln(x + 2) + 12 ln(x + 4)

21) 14ln(x + 2) + 3 ln(x + 4)

22) ln(x + 2) - 12 ln(x + 4)

23) -2 ln(x + 2) - 3 ln(x + 4)24) ln(x + 1) - 2 ln x - ln(x - 3)

25) ln x + 13ln(x - 3) + 2 ln(x + 1)

26) ln x + 2 ln(x - 3) - 12 ln(x + 1)

27) ln 94

28) ln 46 = ln 2 6

3

Page 195

29) ln 94

30) ln(ex)

31) log x2

(x + 7)3

32) 033) 234) 235) 536) 1

37) 167

38) 43

39) ln xln 6

40) ln(x + 3)ln 10

41) log(x + 4)log 2

42) C43) E44) B45) B46) C47) B48) C49) B50) B51) B52) E53) C54) B55) A56) D57) D58) E59) C60) A61) 3.2462) 1663) 78.764) -36.165) x = 266) x5

67) x2

y3

68) 1186.2 million69) 77 years70) no solution

Page 196

71) log(300,000) - log(150,000) = log 300,000150,000

= log(2) ≈ 0.372) log(37,000) + log(1,000) = log(37,000 · 1,000)

= log(37,000,000)≈ 7.568

73) log(250,000) + log(2) = log(250,000 · 2)= log(500,000)≈ 5.7

74) log(10x+2) = x + 275) log 103(x-1) = 3(x -1)76) log(2x · 5x) = log(10x) = x

77) log(36)log(3)

≈ 3.26

78) log(36)log(6)

= 2

79) log(36)log(18)

≈ 1.24

80) log(36)log(9)

≈ 1.63

81) log(36)log(362)

= 0.5

82) y = log(x)log(5)

; y = ln(x)ln (5)

83) y = log(x)log(8)

; y = ln(x)ln(8)

84) y = log(x)log(0.5)

; y = ln(x)ln(0.5)

85) y = log(x)log(12)

; y = ln(x)ln(12)

86) The graph of f(x) = log3 x

x-10 -5 5 10

y10

5

-5

-10

(1/9, -2)(1/3, -1)

(1, 0)(3, 1)

(9, 2)

x-10 -5 5 10

y10

5

-5

-10

(1/9, -2)(1/3, -1)

(1, 0)(3, 1)

(9, 2)

Page 197

87) Thegraph of f(x) = -3log2 (x + 4) + 1

x-10 -5 5 10

y10

5

-5

-10

(-3, 1)

(-2, -2)(0, -5)

(4, -8)

x-10 -5 5 10

y10

5

-5

-10

(-3, 1)

(-2, -2)(0, -5)

(4, -8)

4.4 Logarithmic and Exponential Equations1) log3

log4

2) 3 + log 42

3) - ln 11ln 2

4) 35) 48

6) 14

7) 12

8) 1

9) 73

10) 27

11) 6

12) q = 20 - log plog 3

13) - 92

14) log(1000 - q)log 3

15) D16) A17) C18) B19) B20) 121) x = .6443547

22) x = 75 = 1.4

23) x = ln 3ln 2

= 1.5849625

24) -.648636725) 3

Page 198

26) 8 days27) 1228) 4 days29) a little over 18 years30) Day 1531) a little over 17.2 years32) $20.12; 8 units33) $40.25; 10 units34) $60.00; 27 units35) $20.00; 5 units36) The other earthquake is 2,405,698.7 times as intense as a zero-level earthquake.37) The other earthquake is 1037.5 times as intense as a zero-level earthquake.38) The other earthquake is 28,663,716.3 times as intense as a zero-level earthquake.

Page 199

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Ch. 4 Exponential and Logarithmic Functionsstaff.uob.edu.bh/files/671125478_files/ch04.pdf · Ch. 4 Exponential and Logarithmic Functions 4.1 Exponential Functions SHORT ANSWER. Write