Deck 5: Exponential and Logarithmic Functions

Question 1

The above graph is best represented by

A) y =
B) y =
C) y =
D) y =
E) y = ln 4

Answer 1

y =

Question 2

The population of a city is given by P = 10,000(1.0
where t is the number of years after 1988.Find the population in
(a)1988
(b)1989
(c)1990.

Answer 2

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

Question 3

Graph y = f(x)=
.

Answer 3

Question 4

Graph y = f(x)=
.

Question 5

The population of a city is given by P = 1,000,000(1.0 where t is the number of years after 1987.The population in 1989 was

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

Question 6

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.

Question 7

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

Question 8

Of the following the best approximation of e is

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

Question 9

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

Question 10

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

Question 11

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

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

Question 12

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

Question 13

Which of the following is true? If f(x)= ,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.

Question 14

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

Question 15

The population of a city is given by P = 100,000(1.0
where t is the number of years after 1988.Find the population in
(a)1988
(b)1989
(c)1990.

Question 16

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

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

Question 17

Suppose that the number of patients admitted into a hospital emergency room during a certain hour of the day has a Poisson distribution with mean 3.Find the probability that during that hour there will be exactly two emergency patients.Assume that
= 0.05.

Question 18

A radioactive element is such that N grams remain after t hours,where
How many grams remain after 30 hours?

Question 19

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?

Question 20

For the function f(x)=

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

Question 21

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

Question 22

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

Question 23

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 an expression 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.
(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.

Question 24

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.

Question 25

Sean and Carley have bacterial infections.Sean was given medicine which reduces the number of bacteria hourly by 20%.5 hours later Carley began the same treatment.
(a)If y = 0.
represents the multiplicative decrease of bacteria for Carley,write an equation using the same reference 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 graph the multiplicative decrease of the bacteria in Sean? Verify your answer using a graphics calculator.

Question 26

Assume the value of a sailboat depreciates by
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 an expression 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.
(d)Use your table to guess when your boat will be worth 25% as much as its original price.

Question 27

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

Question 28

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.

Question 29

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

Question 30

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.

Question 31

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)?

Question 32

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.

Question 33

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
% compounded quarterly,how much money should be paid initially into the trust fund?

Question 34

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

Question 35

By looking at the graph of y =
,sketch a graph of y =
- 3.

Question 36

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 an expression 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.
(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.

Question 37

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

Question 38

Assume your savings consist of a $5000 investment which is guaranteed to increase by 7% every year and $800 cash 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 graphing calculator.

Question 39

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

Question 40

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

Question 41

Express

= -3 in exponential form.

Question 42

The number of yearly visitors to a resort has been shrinking at the rate of 6%.Currently the resort gets 120,000 tourists 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 than 20,000.

Question 43

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.

Question 44

Evaluate and simplify: ln

Question 45

Graph y = f(x)=
x.

Question 46

Evaluate and simplify: log 100

Question 47

Express
= 81 in logarithmic form.

Question 48

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 its current size.

Question 49

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.

Question 50

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

Question 51

Express
64 = 3 in exponential form.

Question 52

Graph f(x)=

Question 53

Express
=
in logarithmic form.

Question 54

Graph g(x)=

Question 55

The multiplicative decrease in purchasing power P after t years of inflation at 3% can be modeled by
Graph the decrease in purchasing power as a function of t years.

Question 56

Graph f(x)=
- 2

Question 57

Graph f(x)= 3

Question 58

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

Question 59

Graph h(x)=
- 1

Question 60

Graph the population for a city with 80,000 people as a function of years if the growth is modeled by

Question 61

Find x: ln x = -2

Question 62

Find x: ln
= x

Question 63

Evaluate and simplify:
1

Question 64

Find x:
x = -3

Question 65

Evaluate and simplify:
(-2)

Question 66

Evaluate and simplify:
6

Question 67

Evaluate and simplify: ln

Question 68

Find x: logx = 0

Question 69

Find x: ln x = 1

Question 70

Find x: log 0.01 = x

Question 71

Evaluate and simplify: log (0.1)

Question 72

Find x:
36 = x

Question 73

Find x: log 100,000 = x

Question 74

Find x:
x = 2

Question 75

Evaluate and simplify:

Question 76

Find x: log x = -2

Question 77

Find x:
x = 3

Question 78

Find x:

= x

Question 79

Find x: log x = 3

Question 80

Find x:
2 = x