85)
Evaluate and simplify: 10log 4
85)
86)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln x+ 2
x+ 4
86)
87)
Assume ln x= 2; ln y= 7. Find ln(xy2).
87)
88)
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.
88)
89)
Simplify: e2 ln x– 3 ln y
89)
90)
Write the following in terms of ln x, ln(x– 3), and ln(x+ 1): ln x3(x– 3)(x+1)2
90)
91)
Find x: logx(4x–3) = 2
91)
92)
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.
92)
93)
For the function f(x) =1
3
x
,
(a) what is the domain of f?
(b) What is the range of f?
93)
94)
Express log21
8= – 3 in exponential form.
94)
95)
The population in the state of California is 24 million today. It is increasing at a rate of 0.9%
per year. The population t years from now is given by the rule P= 24e.009t. After how
many years will the population double?
95)
96)
What power of 362 is 36?
96)
97)
Find x: logx= 0
97)
98)
Given ln x= 7.1; ln y= 8.2, find ln(x3y7)
98)
99)
If log 4 = 0.6 and log 7 = 0.8, find log4 7.
99)
22
100)
Solve for x: log(98 –x+x2) = 2
100)
101)
Find x: log3x= 3
101)
102)
Your friend started a savings plan with 3 pennies and each day she saved three times the
amount she saved the previous day. Later you started a savings plan with 9 pennies and
each day you saved 9 times the amount you saved the previous day. On the day that your
friend had been saving for 8 less than 4 times as many days as you, you saved the same
amount as your friend. How many days have you been saving?
102)
103)
A manufacturer’s supply equation is p= log 2 +3q
5 where q is the number of units
supplied at price p per unit. At what price will the manufacturer supply 200 units?
103)
104)
Write the following in terms of ln x, ln(x– 3), and ln(x+ 1): ln x+ 1
x2(x– 3)
104)
105)
The demand equation for a product is given by q= 1000 –3p. Solve for p and express your
answer in terms of common logarithms.
105)
106)
106)
23
107)
The population of a city is given by P= 100,000(1.03)t where t is the number of years after
1988. Find the population in
(a) 1988
(b) 1989
(c) 1990.
107)
108)
Solve for x: eln(4x)= 20
108)
109)
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= 0.8t 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.
109)
110)
If an earthquake is 2x· 5x times as intense as a zero–level earthquake, what is its
measurement on the Richter Scale? Write as a logarithm expression and simplify.
110)
111)
Assume that log 3 = 0.4771 and log 4 = 0.6021. Determine the value of log 36.
111)
112)
Find x: log51
25 =x
112)
113)
True or False: If logax=logbx, then a=b.
113)
114)
The value of an investment of $1000 earning 8% compounded yearly is given by A=
1000(1.08)t, where t is the number of years it has been invested .If the amount of your
investment is now $4000, how long has it been invested?
114)
115)
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.
115)
116)
A radioactive element is such that N grams remain after t hours, where N= 20e–0.028t.
How many grams remain after 30 hours?
116)
117)
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.
117)
118)
Suppose an investment increases by 10% every year. . Graph the number of years invested
as a function of the multiplicative increase in original investment. Label the graph with the
name of the function.
118)
119)
Assume that log 5 = 0.6690 and log 6 = 0.7782. Determine the value of log 6
5.
119)
120)
An earthquake measuring 5.8 on the Richter scale can be represented by 5.8 = log I
I0
where I is the intensity of the earthquake and I0 is the intensity of a zero–level earthquake.
Represent this equation in exponential form.
120)
26
121)
Find x: ln e3=x
121)
122)
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.
122)
123)
The value of an investment of $3000 earning 7.25% compounded yearly is given by A=
3000(1.0725)t, where t is the number of years it has been invested. If the amount of your
investment is now $10,000, how long has it been invested?
123)
124)
Assume that log 5 = 0.6690. Determine the value of log 500.
124)
125)
Find x: log4x= 2
125)
126)
Solve for x: loge2x= 5
126)
127)
Hannah wants to use her graphing calculator to check this sketch of y=log12 x but her
calculator does not make log12 calculations. Find two equations she could use. Use a
graphing calculator to confirm that the two equations are equivalent.
127)
128)
If ln 2 = 0.7 and ln 5 = 1.6, find log2 5.
128)
27
129)
Brett wants to use his graphing calculator to check his sketch of y=log0.5 x but his
calculator does not make log0.5 calculations. Find two equations he could use. Use a
graphing calculator to confirm that the two equations are equivalent.
129)
130)
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 3
4% compounded quarterly, how much money
should be paid initially into the trust fund?
130)
131)
Suppose an investment triples every decade. Graph the number of decades invested as a
function of the multiplicative increase in original investment. Label the graph with the
name of the function.
131)
132)
Solve for x: log2(x– 4) +log2 3 =log2x
132)
133)
April took a number and multiplied it by a power of 9. Bob started with the same number
and got the same result 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?
133)
134)
What power of 18 is 36?
134)
135)
Express ln 4 –1
2(ln 2 + ln 3) as a single logarithm.
135)
136)
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?
136)
137)
Solve the equation for x in terms of y: 1.2y= log x
1.2 ×105
137)
138)
By looking at the graph of y=ex, sketch a graph of y=ex+2– 3.
138)
139)
Solve for x: log(x+ 1) – log(x– 2) = 1
139)
140)
Find x: logx(4x– 1) = 1
140)
29
141)
Graph y=f(x) =(0.5)x.
141)
142)
Consider g(x) = – log3(x + 2)
(a) Graph g(x)
(b) Find the domain of g(x)
142)
143)
Evaluate and simplify: log (0.1)
143)
144)
The magnitude (Richter Scale) of an earthquake is given by R= log I
I0 where I is the
intensity of the earthquake and I0 is the intensity of a zero–level reference earthquake. I
I0
represents how many times greater than the earthquake is than the reference earthquake. If
an earthquake measured 4.2 on the Richter Scale, how many more times intense is it than a
zero–level earthquake?
144)
145)
Assume that log 3 = 0.4771. Determine the value of log 27.
145)
146)
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.
146)
147)
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 t represent?
147)
148)
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 1
h. Represent this equation in exponential form.
148)
149)
Evaluate and simplify: log2(–2)
149)
150)
Write log(x+ 3) in terms of natural logarithms.
150)
151)
Graph y=f(x) =log4x.
151)
152)
Assume that log 6 = 0.7782. Determine the value of log 36.
152)
153)
Graph h(x) = – 2x+3 – 1
153)
154)
Evaluate and simplify: log71
154)
155)
Assume that log 3 = 0.4771. Determine the value of log (0.09).
155)
156)
Solve for x: logx 3 =1
2
156)
157)
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.
157)
158)
The magnitude (Richter Scale) of an earthquake is given by R= log I
I0 where I is the
intensity of the earthquake and I0 is the intensity of a zero–level reference earthquake. I
I0
represents how many times greater the earthquake is than the reference earthquake. Find
the magnitude of an earthquake that is 200 times the intensity of a zero–level earthquake.
158)
159)
Express 2 log(x) – 3 log(x+ 7) as a single logarithm.
159)
160)
The demand function for a product is p= 90 5–q/10 where q is the number of units and p is
the price of one unit. At what price will the demand be 5 units? How many units will be
demanded if the price is $18.00?
160)
161)
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.
161)
162)
Suppose a garbage company has found that garbage per family has decreased by 10%
every year since the first year they started a curb–side recycling program. Graph each year
as a function of the multiplicative decrease in garbage since the first year of the curb–side
recycling program. Label the graph with the name of the function.
162)
163)
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.
163)
164)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln 4(x+ 2)(x+4)3
164)
165)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln 1
(x+2)2(x+4)3
165)
36
166)
Suppose a computer decreases in value by 50% every year. Graph the number of years it is
owned as a function of the multiplicative decrease in its original value. Label the graph
with the name of the function.
166)
167)
Express 34= 81 in logarithmic form.
167)
168)
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.
168)
169)
Evaluate and simplify: log31
81
169)
170)
Solve for x: log(x2+ 4x+ 104) = 2
170)
37
171)
Write log6x in terms of natural logarithms.
171)
172)
Solve for x: logxy= 3
172)
173)
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 =7
8
t
. Represent this equation in logarithmic
form.
173)
174)
Assume that log 3 = 0.4771 and log 4 = 0.6021. Determine the value of log 12.
174)
175)
Solve for x: ln x+ ln 3 = ln(x+ 1)
175)
176)
Find x and express your answer in terms of natural logarithms: 2–x– 3 = 8
176)
177)
Solve for x: ln(x+ 3) = ln(2x)
177)
178)
What is the sum of the Richter Scale measurement of an earthquake which is 37,000 times
the intensity of a zero–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 to evaluate.
178)
179)
Evaluate and simplify: log 100
179)
180)
Express 2(ln 4 + ln 3 – 3 ln 2) as a single logarithm.
180)
181)
Solve for x: 2log2x+log25= 7
181)
182)
Assume log3x= 2; log3y= .12. Find log33xy2.
182)
183)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln (x+ 2) x+ 4
183)
184)
Assume the value of a sailboat depreciates by 1
8 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.
184)
39