186)
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.
186)
187)
Assume that log 3 = 0.4771 and log 4 = 0.6021. Determine the value of log 36.
187)
188)
Solve for x: 22x– 2 ·2x– 3 = 0
188)
189)
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.
189)
41
190)
Find x and express your answer in terms of common logarithms: 102x–3= 4
190)
191)
If $400 is invested for 2 years at 6% compounded semiannually, find
(a) the compound amount and
(b) the compound interest.
191)
192)
For the function f(x) =log4x,
(a) what is the domain of f?
(b) What is the range of f?
192)
193)
Find x: logx(4x– 1) = 1
193)
194)
194)
42
43
195)
Use the change base formula and your graphing calculator to graph
f(x) =log3x
195)
196)
Evaluate and simplify: log71
196)
197)
True or False: If logax=logbx, then a=b.
197)
198)
Solve for x: log(x2+ 4x+ 104) = 2
198)
199)
Evaluate and simplify: log5
352
199)
44
200)
Graph the population for a city with 80,000 people as a function of years if the growth is
modeled by P= 80,000e0.04t.
200)
201)
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?
201)
202)
Find x: log4 2 =x
202)
203)
Find x: log4x= 2
203)
45
204)
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.
204)
Answer:
205)
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.
205)
Answer:
206)
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.
206)
Answer:
46
207)
R= log I
I0 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 a zero–level earthquake to the Richter Scale measurement.
207)
208)
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.
208)
209)
Find x and express your answer in terms of natural logarithms: 2–x– 3 = 8
209)
210)
Assume that log 3 = 0.4771 and log 4 = 0.6021. Determine the value of log 12.
210)
47
211)
The demand function for a product is p= 180 3–q/15 where q is the number of units and p
is the price of one unit. At what price will the demand be 15 units? How many units will be
demanded if the price is $24.91?
211)
212)
What power of 362 is 36?
212)
213)
Your friend started a savings plan with 2 pennies and each day he doubled the amount
saved. Later you started a savings plan with 4 pennies and each day you quadrupled the
amount saved. Your friend has been saving for 6 less than 3 times as many days as you. In
two days you will put the same amount in savings as your friend. How many days have
you been saving?
213)
214)
Solve: 10log x3= 27
214)
215)
A radioactive substance decays according to the equation N=10e–0.04t, where N is the
number of milligrams present after t days. Find the half–life of the substance.
215)
216)
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.
216)
217)
Solve the equation for x in terms of y: 1.2y= log x
1.2 ×105
217)
218)
Solve for x: ln e2x=x
218)
48
219)
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?
219)
220)
Solve for x: (2 +x)5= 129.3
220)
221)
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.
221)
49
222)
Graph f(x) =2x
222)
223)
Assume that log 5 = 0.6690. Determine the value of log 500.
223)
224)
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?
224)
225)
Find x: logx16 = 4
225)
226)
Evaluate and simplify: log6 6
226)
50
227)
Find x: log6 36 =x
227)
228)
Solve for x: 4x+1=83x
228)
229)
The multiplicative increase m of an investment which is invested at an annual rate of p and
compounded continuously for a time t is given by m=ept. If your annual rate is 6.75%, how
many years will it take to double your investment?
229)
230)
Write the following in terms of ln x, ln(x– 3), and ln(x+ 1): ln x3(x– 3)(x+1)2
230)
231)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln (x+ 2) x+ 4
231)
232)
Find x: logx= 0
232)
233)
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.
233)
234)
Evaluate and simplify: log2(–2)
234)
235)
Assume that log 6 = 0.7782. Determine the value of log 36.
235)
51
236)
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.
236)
237)
The demand function for a product is p= 60 3–q/15 where q is the number of units and p is
the price of one unit. At what price will the demand be 15 units? How many units will be
demanded if the price is $41.60?
237)
238)
If an earthquake is 10x+2 times as intense as a zero–level earthquake, what is its
measurement on the Richter Scale? Write as logarithmic expression and simplify.
238)
239)
Evaluate and simplify: 10log 4
239)
240)
Express 2 log(x) – 3 log(x+ 7) as a single logarithm.
240)
241)
The sales manager of a department store finds that its daily sales begins to fall after the end
of a promotional campaign. The sales in dollars as a function of the number of days after
the campaign’s end is given by S(d) = 36,000 9
8
–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?
241)
242)
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.
242)
52
243)
Evaluate and simplify: log31
81
243)
244)
Evaluate and simplify: log2 1 +log2 2
244)
245)
Consider g(x) = – log3(x + 2)
(a) Graph g(x)
(b) Find the domain of g(x)
245)
246)
How long will it take for $100 to amount to $200 at an interest rate of 10% compounded
annually? Give your answer to 2 decimal places.
246)
247)
Evaluate and simplify: ln e
247)
53
248)
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?
248)
249)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln (x+2)2(x+ 4)
249)
250)
By looking at the graph of y=ex, sketch a graph of y=ex+2– 3.
250)
251)
If an earthquake is 103(x–1) times as intense as a zero–level earthquake, what is its
measurement on the Richter Scale? Write as logarithmic expression and simplify.
251)
252)
Find x: ln x= 1
252)
253)
Find x: log446=x
253)
254)
Use a graphing calculator to approximate the solution 2x–3x= 20.
254)
54
255)
Graph y=f(x) =3x.
255)
256)
Assume ln x= 2; ln y= 7. Find ln(xy2).
256)
Answer:
16
Explanation:
257)
Find x: log2x= – 3
257)
Answer:
Explanation:
258)
For the function f(x) =5x,
(a) what is the domain of f?
(b) what is the range of f?
258)
Answer:
Explanation:
55
Answer:
Explanation:
Answer Key
Testname: C4
56