91)
Solve for x: 2log2x+log25= 7
91)
92)
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)?
92)
93)
Steve wants to use his graphing calculator to check his sketch of y=log5x but his
calculator does not make log5 calculations. Find two equations he could use.
93)
94)
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.
94)
95)
If ln 2 = 0.7 and ln 5 = 1.6, find log2 5.
95)
96)
Find x if 7e3x= 1
96)
97)
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.
97)
98)
Solve for x: log(x+ 1) – log(x– 2) = 1
98)
99)
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 2,000,000 times the intensity of a zero–level
earthquake.
99)
100)
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.
100)
101)
Evaluate and simplify: log 100
101)
102)
What power of 9 is 36?
102)
103)
Solve for t: 300 = 500(1 –e0.2t). Assume that ln(0.4) = – 0.9.
103)
104)
Write log(x+ 3) in terms of natural logarithms.
104)
23
105)
If log 4 = 0.6 and log 7 = 0.8, find log4 7.
105)
106)
The population of India was 651 million in 1980 and has been growing at a rate of 2% per
year. The population t years later is approximated by N(t) =651e.02t. Estimate the
population in India in the year 2010.
106)
107)
Solve for x: logx(2x+ 3) = 2
107)
108)
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?
108)
109)
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?
109)
110)
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(
ft). 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 f= 1. How does the number of bacteria change when f= 1?
110)
111)
Find x: ln x= – 2
111)
112)
Write log6x in terms of natural logarithms.
112)
113)
Express 1 + ln x as a single logarithm.
113)
114)
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.
114)
Hours Bacteria Expression
0500 500(1.4)0
1700 500(1.4)1
115)
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.
115)
116)
An earthquake which is 520,000 times as intense as a zero–level earthquake has a
magnitude on the Richter Scale which is 2.7 more than the intensity of another earthquake.
What is the intensity of the other earthquake?
116)
117)
Express log4 64 = 3 in exponential form.
117)
118)
Find x and express your answer in terms of natural logarithms: e4x= 2
118)
119)
Find x: logx(4x–3) = 2
119)
120)
Graph h(x) = – 2x+3 – 1
120)
121)
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?
121)
122)
Assume that log 3 = 0.4771. Determine the value of log (0.09).
122)
123)
Solve for x: eln(3x+4) = 10
123)
124)
What is the sum of the Richter Scale measurement of an earthquake which is 250,000 times
the intensity of a zero–level earthquake and an earthquake with intensity twice the
intensity of a zero–level earthquake? Write as an expression involving logarithms.
Simplify by combining logarithms and then use a calculator to evaluate.
124)
125)
Assume that log 4 = 0.6021. Determine the value of log 400.
125)
126)
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.
126)
Hours Bacteria Expression
0128 128 · 20
1256 128 · 21
127)
Find x: log3x= 3
127)
128)
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.
128)
27
129)
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?
129)
130)
Write log2(x+ 4) in terms of common logarithms.
130)
131)
Express 10–3=1
1000 in logarithmic form.
131)
132)
For the function f(x) =1
3
x
,
(a) what is the domain of f?
(b) What is the range of f?
132)
133)
Solve for x: alogax+loga4= 8
133)
134)
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.
134)
135)
Find x: log 0.01 =x
135)
136)
Solve for x: logx 3 =1
2
136)
137)
Assume that log 3 = 0.4771. Determine the value of log 27.
137)
138)
What power of 6 is 36?
138)
139)
Graph y=f(x) =log4x.
139)
140)
Assume that log 4 = 0.6021. Determine the value of log 1
16 .
140)
141)
Solve for x: e2ln(2x)= 4
141)
142)
If the yeast has been decreasing by 80% hourly and the current amount is 1
78,125 of the
amount first measured, then the situation can be represented by 1
78,125 =1
5
t
. Represent
this equation in logarithmic form. What does t represent?
142)
143)
How much more on the Richter Scale is an earthquake with intensity 300,000 times the
intensity of a zero–level earthquake than an earthquake with intensity 150,000 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.
143)
144)
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?
144)
145)
Solve for x: ln(x+ 1) – ln x= ln 2
145)
146)
An earthquake which is 48,000 times as intense as a zero–level earthquake has a
magnitude on the Richter Scale which is 1.7 less than the intensity of another earthquake.
What is the intensity of the other earthquake?
146)
147)
Find x: log51
25 =x
147)
148)
Solve for x: 42x= 2
148)
149)
Solve for x: ln x+ ln 3 = ln(x+ 1)
149)
150)
Find the domain of y=log3(x+ 1)
150)
31
151)
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.
151)
152)
Simplify: 105log x
152)
153)
Solve for x: ln(x+ 3) = ln(2x)
153)
154)
Solve for x: loge2x= 5
154)
155)
Assume that log 5 = 0.6690. Determine the value of log 35.
155)
156)
Solve for x: eln(4x)= 20
156)
157)
Evaluate and simplify: log (0.1)
157)
158)
Assume that log 3 = 0.4771 and log 4 = 0.6021. Determine the value of log 4
3.
158)
159)
Solve for x: 3log3x+log34= 8
159)
32
160)
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.
160)
161)
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?
161)
162)
Write the following in terms of ln x, ln(x– 3), and ln(x+ 1): ln x+ 1
x2(x– 3)
162)
163)
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.
163)
164)
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.
164)
165)
Use the change base formula and your graphing calculator to graph
f(x) = – 3log2 (x + 4) + 1
165)
166)
What power of 3 is 36?
166)
167)
A radioactive element is such that N grams remain after t hours, where N= 20e–0.028t.
How many grams remain after 30 hours?
167)
168)
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.
168)
169)
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.
169)
170)
Express 2(ln 4 + ln 3 – 3 ln 2) as a single logarithm.
170)
35
171)
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 7.5 on the Richter Scale, how many more times intense is it than a
barely–felt earthquake?
171)
172)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln x+ 2
x+ 4
172)
173)
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.
173)
174)
What power of 18 is 36?
174)
175)
Graph f(x) =ex– 2
175)
176)
Express 2 ln 3 – ln 4 as a single logarithm.
176)
177)
Find x: log 100,000 =x
177)
178)
178)
37
179)
Express ln 4 –1
2(ln 2 + ln 3) as a single logarithm.
179)
38
180)
Solve for x: logxy= 3
180)
181)
Graph f(x) = 3 2
3
x
181)
x
182)
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.
182)
183)
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.
183)
184)
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.
184)
185)
April wants to use her graphing calculator to check her sketch of y=log8x but her
calculator does not make log8 calculations. Find two equations she could use.
185)