Chapter 5
Exponential and Logarithmic Functions
Exercise 5A
1. Expand e  8 x in ascending powers of x up to x 3.
2. Expand e2 x  e 2 x in ascending powers of x up to x 4 .
3. Expand 2e2 x  4e x in ascending powers of x up to x 3.
1
4. Expand
e
5. Simplify
6. Simplify
x
in ascending powers of x up to x 4 .
(e 2 ) x
3
e4 x
.
e3 x  2e5 x
2  2e

.
x
2
7. Find lim [ e2 x  ln(x  2)] . (Give your answer correct to 4 significant figures.)
x 0
8. Find lim [(3x  1)(5  2 x )] .
2
x  
9. Find lim ln(e 4 
x 
3 6
) .
5e x
10. Find lim
5e  0.1x  3
.
2e  0.1x  5
11. Find lim
e4 x
.
e2 x  e 2 x
x 
x  
12. Find lim 6 ln(e 2 
x 
7
).
3  ex
2
13. Solve the equation e x  3. (Give your answers correct to 4 significant figures.)
2
14. Solve the equation e x  e2 x  e3 .
15. Solve the equation e3  ln( x 1)  3. (Give your answer correct to 4 significant figures.)
16. Solve the equation e2 ln x  2 x 2  4 x  5 .
17. Solve the equation 3x  e3 . (Give your answer correct to 4 significant figures.)
18. Solve the equation e x 1  3x 1. (Give your answer correct to 4 significant figures.)
19. Solve the equation 12e2 x  1  7e x . (Give your answers correct to 4 significant figures.)
20. Solve the equation 2e x  11  21e x  0. (Give your answer correct to 4 significant figures.)
21. Solve the equation ln(5x  4e2 )  2 . (Give your answer correct to 4 significant figures.)
22. Solve the equation ln(x  1)  ln(x  2)  ln 4.
23. Solve the equation [ln(x  1)]2  2 ln(x 2  2 x  1) . (Give your answers correct to 4 significant figures if
necessary.)
P. 1
24. Expand
e 2 x  e x
in ascending powers of x up to x 3.
x
xe
25. Expand (2 x  1)e x in ascending powers of x up to x 2 .
26. Expand e3 x  e3 x in ascending powers of x up to x 4 .
2
27. Expand e x
28. Expand
2
2x
x2
e3
x
2
e
in ascending powers of x up to x 4 .
in ascending powers of x up to x 4 .
29. (a) Expand e  3 x in ascending powers of x up to x 3.
(b) Expand (3x  2)6 in ascending powers of x up to x 3.
(c) Hence expand
(3x  2)6
in ascending powers of x up to x 3.
3x
e
30. (a) Prove that 3x  e x ln 3.
(b) Hence expand 3 x in ascending powers of x up to x 3.
31. (a) Prove that 6 x  e x ln 6.
(b) Expand ( x  2) 4 .
(c) Hence expand 6 x ( x  2) 4 in ascending powers of x up to x 3.
1
32. It is given that f ( x)  ka x , where a and k are constants. If f ( )   4 and f (1)  108, find the values of a
2
and k.
33. It is given that f ( x)  Aekx, where A and k are constants. If f (1)  140.6 and f (2)  2 824, find the values
of A and k. (Give your answers correct to 1 significant figure.)
34. It is given that T  T0e kt , where T0 and k are constants. T  895 when t  2 ; T  298 when t  3.5. Find
the values of T0 and k. (Give your answers correct to 1 significant figure.)
2
35. It is given that y 
, where a and k are constants. y  0.5 when t  0 ; y  0.428 8 when t  1. Find
1  aekt
the values of a and k. (Give your answers correct to 1 significant figure if necessary.)
e x 1  32 x 5 . (Give your answer correct to 4 significant figures.)
36. Solve the equation
7
37. Solve the equation
4 x 5
3 2 x
(Give your answer correct to 4 significant figures.)
x 1
 15. (Give your answer correct to 4 significant figures.)
x
y
2e  e  10
39. Solve the simultaneous equations 
. (Give your answers correct to 4 significant figures.)
e x  2e y  5
38. Solve the equation 5
e

x2
e 4 .
x
 e
40. Solve the simultaneous equations  y
. (Give your answers correct to 4 significant figures.)
2 ln x  ln y  1

P. 2
41. After charging a rechargeable battery for t hours, the electrical energy L units stored in the battery can be
modelled by L  1 000  ae kt where a and k are constants. It is given that the initial electrical energy and
the electrical energy after charging for 5 hours are 200 units and 892 units respectively.
(a) Find the values of a and k.
(b) When the battery has charged for a long period of time, what is the electrical energy stored in the
battery?
(Give your answers correct to 2 significant figures if necessary.)
42. After t days of the outbreak of the foot-and-mouth disease, the accumulated number of infected pigs N can
4 800
be modelled by N 
, where a and b are constants. It is known that the initial number of infected
6  aebt
pigs is 80 and the accumulated number of infected pigs is 460 after 5 days. Find the values of a and b.
(Give your answers correct to 1 significant figure if necessary.)
a
, where a and b are constants. y  15.09 when x  4 ; y  61.46 when x  2 .
1  2ebx
Find the values of a and b. (Give your answers correct to 1 significant figure.)
43. It is given that y 
44. Solve the equation e 2 x 
e 4 x  2e 2
. (Give your answers correct to 4 significant figures.)
8
45. Solve the equation 3e2  4e2 x 1  e4 x  0. (Give your answers correct to 4 significant figures if necessary.)
46. The sea is polluted by the chemical spill after an accident of a cargo ship. It is given that the concentration
of the chemical C mg/L at d km away from the accident spot is given by C  C0e kd , where C0 and k are
constants and k  0 . A research finds that the concentrations of the chemical in the sea at 50 km and
100 km away from the accident spot are 400 mg/L and 190 mg/L respectively.
(a) Find the concentration of the chemical at the accident spot.
(b) If the safety standard of the concentration of the chemical is 200 mg/L, find the area of the sea which
exceeds the safety standard.
(Give your answers correct to 2 significant figures.)
47. After cooling a cup of tea for t minutes, its temperature TC can be modelled by T (t ) 
750e 0.08t  160
5e 0.08t  8
.
(a) Find T(0) and interpret the meaning of T(0).
(b) At least how long will it take for the temperature of the cup of tea to be lower than 55C? (Give your
answer correct to 3 significant figures.)
(c) After cooling the cup of tea for a long period of time, what will be its temperature?
48. An athlete takes a rest after practising for 30 minutes. A research finds that the concentration of lactic acid
in the bloodstream C(t) (in mM) after the athlete has practised for t minutes can be modelled
by C (t )  a  0.04(t 2  9t  6)e bt , where a and b are positive constants.
(a) Interpret the meaning of lim C (t ).
t 
(b) It is given that lim C (t )  1.5, find the value of a.
t 
[ Hint: limt n e  t  0 , where n and  are positive constants. ]
t 
(c) It is given that the concentration of lactic acid in the bloodstream of the athlete after practi sing for 30
minutes is 3.58 mM higher than that after practising for 15 minutes. Find the value of b. (Give your
answer correct to 3 significant figures.)
P. 3
Exercise 5B
49. A principal of $10 000 is deposited in a bank at an interest rate of 4 p.a. According to each of the
following conditions, find the amount after 3 years.
(a) The interest is compounded monthly.
(b) The interest is compounded continuously.
(Give your answers correct to the nearest dollar.)
50. A principal $P is deposited in a bank at an interest rate of r% p.a. compounded continuously. If the
interest after 20 years is two times of the principal, find the interest rate p.a. (Give your answer correct to
3 significant figures.)
51. An insurance company provides a savings plan of which the interest on investment is compounded
continuously. The company guarantees that the amount in the plan triples the original investment for every
12 years.
(a) Find the interest rate p.a. provided by the insurance company.
(b) If $10 000 has been invested in the plan originally, how long will it take for the amount in the plan to
increase to $50 000?
(Give your answers correct to 3 significant figures.)
52. The population of a small town increases at a rate of 1% per year continuously. If the population is 15 000
at present, find the population of the small town 6 years ago. (Give your answer correct to 3
significant figures.)
53. Mr. Wong bought a car for $250 000. The value of the car depreciates at a constant rate of 20% per year
continuously.
(a) Find the value of the car after 4 years.
(b) It is given that Mr. Wong will buy a new car when the value of the car depreciates to one -fifth of the
original price. When will he buy a new car?
(Give your answers correct to 3 significant figures.)
54. It is given that the value of a mobile phone 3 months ago was $4 880, and its current value is $4 440. If the
value of the mobile phone decreases at a constant rate continuously, find the rate of decrease pe r month.
(Give your answer correct to 3 significant figures.)
55. The number of undecayed nuclei of a radioactive substance after t years is given by N  Ae t , where the
positive constant  is the decay constant. It is given that it takes 1 000 years for the number of undecayed
nuclei of the substance to reduce to one-tenth of the original number. Find the decay constant of the
substance. (Give your answer correct to 2 significant figures.)
56. (a) The value of an antique increases at a constant rate continuously. If the value of the antique after 10
years will be 30% higher than its original price, find the rate of increase of the antique per yea r.
(b) How long will it take for the value of the antique to be 50% higher than its original price?
(Give your answers correct to 3 significant figures.)
P. 4
57. The value of a property was 2.8 million dollars at the beginning of 2000 and 3.9 million dollars at the end
of 2008. If the value of the property increases at a constant rate continuously, find the rate of increase per
annum. (Give your answer correct to 3 significant figures.)
58. The current number of cars in a small town is 7 250. If the number of cars increases at a constant rate
continuously, the number of cars will increase by 450 after 3 years. Find the rate of increase per annum.
(Give your answer correct to 3 significant figures.)
59. The population of a city is 700 000 at present and the population increased at a constant rate of 0.5% per
quarter over the past 10 years.
(a) Find the population of the city 5 years ago.
(b) If the rate of increase of the population of the city in the coming 10 years will double that in the past
10 years, at least how long will it take for the population of the city to be more than 1 000 000?
(Give your answers correct to 3 significant figures.)
60. 100 mg of medicine is injected into the body of a patient. The amount of the remaining medicine A(t) (in
mg) in the body of the patient t hours after injection can be modelled by A(t )  100e 0.5t .
(a) Find A(0) and interpret the meaning of A(0).
(b) Find the amount of the remaining medicine in the body of the patient 4 hours after injection.
(c) It is given that when the amount of the medicine in the body of the patient is lower than 10 mg, the
efficacy of the medicine will be lost and the patient needs to have another injection. What should be
the interval between two injections?
(Give your answers correct to 3 significant figures if necessary.)
61. A researcher cultured some bacteria in a liquid and found that the number of bacteria present in the liquid
was 700 at 800 a.m. and 19 000 at 1030 a.m. It is given that the number of bacteria present N(t) can be
modelled by N (t )  N0ekt, where t is the time measured in hours and t  0 corresponds to 600 a.m.
(a) Find the value of N 0 and interpret the meaning of N 0 . (Give your answer correct to the nearest
integer.)
(b) When will the number of bacteria reach 500 000? (Give your answer correct to the nearest hour.)
62. The number of undecayed nuclei N of a radioactive substance is given by N  N0e kt , where N 0 is the
number of undecayed nuclei at time t  0, t is the time measured in years and k is a constant. It is given
that the half-life of the substance is 1 200 years.
[ Hint: The half-life of a radioactive substance is the time taken for its number of undecayed nuclei to reduce to half. ]
(a) Find the value of k.
(b) If the initial number of undecayed nuclei is 1 000 000, find the number of undecayed nuclei after
1 800 years.
(c) Find the time required for the number of undecayed nuclei decay from 500 000 to 400 000.
(Give your answers correct to 3 significant figures.)
P. 5
63. The accumulated number of flu cases N recorded in a clinic after t days can be modelled
by N  A  ke 0.01t (0  t  60). It is given that the accumulated number of flu cases after
5 days and
10 days are 50 and 90 respectively.
(a) Find the values of A and k. (Give your answers correct to 3 significant figures.)
(b) What is the accumulated number of flu cases after two weeks? (Give your answer correct to the
nearest integer.)
(c) At least how long will it take for the accumulated number of flu cases be more than 250? (Give your
answer correct to the nearest integer.)
4 1
64. The decay of a radioactive substance can be modelled by the equation M  M 0 (  e  kt ) , where M grams
5 5
is the mass of the substance after t years, M 0 and k are positive constants.
(a) If 85% of the mass of the substance will remain after 700 years, find the value of k. (Give your
answer correct to 3 significant figures.)
(b) How long will it take for the mass to be 95% of the initial mass? (Give your answer correct to 3
significant figures.)
(c) When the substance has decayed for a long period of time, express the mass of the substance in terms
of M 0 .
65. (a) The number of tourists in a European country increases at a constant rate of r% per annum continuously.
It is given that the number of tourists increases by n in t years and the number of tourists is N after
100
n
the increment. Prove that t  
ln(1  ).
r
N
(b) If the rate of increase in (a) is 10% per annum and the number of tourists will increase from 150 000
to 200 000 within a period of time, how long will the period be? (Give your answer correct to 3
significant figures.)
66. Mr. Chan bought a property for 4 million dollars on 1 January 2001. The value of the property P (in
million dollars) can be modelled by P  P0e0.1t , where t (in years) is the time elapsed since the purchase of
the property by Mr. Chan.
(a) Find the value of P0 .
(b) Find the value of the property on 1 January 2005.
(c) Find the percentage change in the value of the property from 1 January 2006 to
2009.
(Give your answers correct to 3 significant figures if necessary. )
1 January
67. When light passes through water, the intensity of light will decrease due to the absorption of light by water.
It is given that the intensity of light I units at the depth x m below the water surface can be modelled
by I  450e 0.21x .
(a) Find the intensity of light at the depth of 5 m below the water surface.
(b) Find the percentage change in the intensity of light from the depth of 5 m to 6 m below the water
surface.
(Give your answers correct to 3 significant figures.)
P. 6
68. The decay of a radioactive substance can be modelled by the equation Q  Q0e kt , where Q is the number of
undecayed nuclei after t years, Q0 and k are positive constants.
(a) If 20% of the undecayed nuclei will remain after 700 years, find the value of k.
(b) If the number of undecayed nuclei after 500 years is 5 000, find the value of Q0 .
(c) Find the percentage change in the number of undecayed nuclei from 100 years later to
later.
(Give your answers correct to 3 significant figures.)
300 years
69. Raymond has joined a savings scheme in a bank to prepare for his study in Australia 4 years later. He will
deposit $P in the bank at the beginning of each of the three consecutive years at an interest rate of 5% p.a.
compounded continuously. He will take out all the money from his account at the beginning of the 4th year
to pay for his tuition fee.
(a) It is given that the required tuition fee is $450 000, find the least value of P.
(b) What is the minimum amount of money to be deposited by him in total?
(c) It is given that he is going to postpone the plan for studying abroad by 1 year and extend the savings
scheme by 1 year, and thus he will deposit $P in the bank at the beginning of each of the four
consecutive years. If the way in calculating the interest does not change, what is the minimum amount
of money to be deposited by him in total?
(Give your answers correct to 4 significant figures.)
70. The number of bicycles of a city increases at a constant rate continuously. It is given that the number of
bicycles after 2 years and 4 years will be greater than the number of bicycles at present by 505 and 1 020
respectively. Find the number of bicycles at present and the rate of increase per annum. (Give your
answers correct to 3 significant figures.)
71. The population of a city increases at a rate of r% per annum continuously. It is given that the population P
(in millions) after t years can be modelled by P  8  20.15t .
(a) Find the initial population.
(b) Find the value of r.
(c) When will the population reach 12 million?
(Give your answers correct to 3 significant figures if necessary. )
72. The sales volume N sets of a video game in the t th month after its launch can be modelled by
N  10 000e 0.2t (t 1) .
(a) Find the sales volume in the 1st month and the 2nd month after the launch. (Give your answers
correct to the nearest integer if necessary.)
(b) Which month has sales volume which is the closest to 500 sets?
P. 7
Exercise 5C
73. The relation of the volume of a sphere V cm3 and its radius R cm is in the form of V 
4 3
R .
3
(a) Express logV in terms of log R .
(b) It is given that the graph of logV against log R is linear, find the slope and the intercept on the
vertical axis of the graph. (Give your answers correct to 3 significant figures if necessary.)
74. The relation of the gas pressure of an experimental instrument P (in atm) and the gas volume V (in m3 ) is
8
in the form of P  .
V
(a) Express ln P in terms of ln V .
(b) It is given that the graph of ln P against ln V is linear, find the slope and the intercept on the vertical
axis of the graph. (Give your answers correct to 3 significant figures if necessary.)
75. Let y  15e

x
5.
(a) Express ln y as a linear function of x.
(b) Write down the slope and the intercept on the vertical axis of the graph of the linear function obtained
in (a). (Give your answers correct to 3 significant figures if necessary.)
76. Let y 
21
.
12  2 4 x
(a) Express ln(
21
 12) as a linear function of x.
y
(b) Write down the slope and intercept on the vertical axis of the graph of the linear function obtained in
(a). (Give your answers correct to 3 significant figures if necessary.)
P. 8
77. The relation of two variables x and y is in the form of y  ka x , where a and k are constants and a  0 . The
corresponding values of x and y are given in the table below.
x
1
3
5
7
9
y
6
54
486
4 374
39 366
(a) Plot the graph of log y against x.
log y
4.5
4
3.5
3
2.5
2
1.5
1
0.5
O
x
1
2
3
4
5
6
7
8
9
10
(b) Use the graph to find the values of a and k. (Give your answers correct to 1 decimal place.)
78. The relation of two variables x and y is in the form of y  ka x , where a and k are constants and a  0 . The
corresponding values of x and y are given in the table below.
x
1
3
5
7
9
y
225.0
126.6
71.2
40.0
22.5
(a) Plot a suitable straight line to find the values of a and k.
ln y
6
5
4
3
2
O
x
2
4
6
8
(b) Find x when y  200.
(Give your answers correct to 2 significant figures.)
P. 9
79. The relation of two variables x and y is in the form of y  ka x , where a and k are constants and a  0 . The
corresponding values of x and y are given in the table below.
x
1
2
3
4
5
y
48.0
153.6
491.5
1 572.9
5 033.2
(a) Plot the graph of ln y against x.
(b) Use the graph to find the values of a and k. (Give your answers correct to 2 significant figures.)
80. The relation of two variables S and t is in the form of S  kt n , where k and n are constants. The
corresponding values of S and t are given in the table below.
t
1
2
3
4
5
S
120.0
169.7
207.8
240.0
268.3
(a) Plot the graph of ln S against ln t .
ln S
6
5.5
5
4.5
O
ln t
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
(b) Use the graph to find the values of k and n. (Give your answers correct to 2 significant figures.)
(c) Find S when t  10. (Give your answer correct to 1 decimal place.)
P. 10
81. The relation of two variables S and t is in the form of S  kt n , where k and n are constants. The
corresponding values of S and t are given in the table below.
t
0.5
1.5
2.5
3.5
4.5
S
155.56
29.94
13.91
8.40
5.76
(a) Plot the graph of ln S against ln t .
(b) Use the graph to find the values of k and n.
(c) Find t when S  100.
(Give your answers correct to 2 significant figures if necessary.)
P. 11
82. The relation of two variables x and y is in the form of y  aekx, where a and k are constants. The
corresponding values of x and y are given in the table below.
x
3
4
5
6
7
y
540.1
2 420.6
10 849.3
48 618.5
217 893.0
(a) Plot the graph of ln y against x.
(b) Use the graph to find the values of a and k. (Give your answers correct to 1 decimal place if
necessary.)
P. 12
83. The relation of two variables x and y is in the form of y  ae kx , where a and k are constants. The
corresponding values of x and y are given in the table below.
x
2
1
4
7
10
y
60.74
38.73
24.70
15.75
10.04
(a) Plot the graph of ln y against x.
(b) Use the graph to find the values of a and k.
(c) Find x when y  10.
(Give your answers correct to 2 significant figures if necessary.)
P. 13
84. The following table shows some corresponding values of x and y.
x
2
4
6
8
10
y
14.14
10.00
8.16
7.07
6.32
(a) Plot the graph of log y against log x .
log y
1.1
1
0.9
0.8
0.7
O
log x
0.2
0.4
0.6
0.8
1
(b) Prove that the relation of x and y is in the form of y  kxn , where k and n are constants.
(c) Hence find the values of k and n. (Give your answers correct to 1 decimal place.)
85. A laptop depreciates at a constant rate continuously and its value $ y after t months is given by y  y0ekt ,
where y0 and k are constants. The corresponding values of t and y are given in the table below.
Time t (month)
3
6
9
12
15
Value y ($)
14 200
12 600
11 200
9 900
8 800
(a) Plot the graph of ln y against t.
(b) Use the graph to find the values of y0 and k. (Give your answers correct to 3 significant figures.)
(c) How long will it take for the value of the laptop to depreciate to 50% of the original
your answer correct to 3 significant figures.)
P. 14
price? (Give
86. The relation of two variables t and y is in the form of y  kat . The corresponding values of t and y are
given in the table below.
t
2
4
6
8
10
y
51.3
320.3
2 002.0
12 512.2
78 201.3
(a) Express ln( y) as a linear function of t.
(b) Plot the graph of ln( y) against t.
(c) Use the graph to find the values of a and k. (Give your answers correct to 1 decimal place.)
P. 15
87. Two quantities P and t are connected by P 
500
, where a and k are constants. The corresponding
1  ae kt
values of P and t are given in the table below.
t
1
2
3
4
5
P
40.3
42.2
44.1
46.2
48.3
(a) Express ln(
500
 1) as a linear function of t.
P
(b) Plot the graph of ln(
500
 1) against t.
P
(c) Use the graph to find the values of a and k. (Give your answers correct to 2 significant figures.)
P. 16
88. Two quantities P and t are connected by P 
3 000
, where a and k are constants. The corresponding
1  aekt
values of P and t are given in the table below.
t
2
4
6
8
10
P
1 722
1 349
994
694
463
(a) Express ln(
3 000
 1) as a linear function of t.
P
(b) Plot the graph of ln(
3 000
 1) against t.
P
(c) Use the graph to find the values of a and k. (Give your answers correct to 2 significant figures.)
(d) Find P when t  20. (Give your answer correct to the nearest integer.)
P. 17
89. Two quantities P and t are connected by P 
150
, where a and k are constants. The corresponding
aekt  1
values of P and t are given in the table below.
t
0.1
0.2
0.3
0.4
0.5
P
0.373 9
0.558 5
0.834 6
1.248 6
1.870 3
150
(a) Express ln(
 1) as a linear function of t.
P
150
(b) Plot the graph of ln(
 1) against t.
P
(c) Use the graph to find the values of a and k. (Give your answers correct to 2 significant figures if
necessary.)
(d) When t tends to infinity, what is the value of P?
P. 18
90. It is given that two quantities x and y are connected by y  ln(ax  b) 2, where a and b are constants. The
corresponding values of x and y are given in the table below.
x
5
10
15
20
25
y
6.59
7.48
8.09
8.55
8.93
(a) Express
y
2
e
as a linear function of x.
y
(b) Plot the graph of e 2 against x.
(c) Use the graph to find the values of a and b. (Give your answers correct to 1 decimal place if
necessary.)
(d) Find y when x  50. (Give your answer correct to 1 decimal place.)
P. 19
91. The harvest y (in kg/ m  ) of a corn field can be modelled by y  500  Aekx, where A and k are constants,
and x is the fertilizer usage (in kg/ m 2 ). The following table shows the details of the harvest of the corn
field obtained in a research.
Usage of fertilizer x (kg/m 2 )
1
2
3
4
5
Harvest y (kg/m 2 )
378.7
426.4
455.4
472.9
483.6
(a) Express ln(500  y) as a linear function of x.
(b) Use the graph to find the values of A and k.
(c) If no fertilizer is used, what is the harvest of the corn field?
(Give your answers correct to 2 significant figures if necessary.)
P. 20
92. The accumulated sales volume N (in thousands) t months after the launch of a domestic game can be
modelled by N  at  b ln t (t  1), where a and b are constants. The following table shows the accumulated
sales volume in the first 15 months of the launch of the domestic game.
Time t (months)
3
6
9
12
15
Accumulated sales volume N
(thousands)
63.9
107.6
136.9
160.2
180.4
(a) Plot the graph of
N
ln t
against
.
t
t
(b) Use the graph to find the values of a and b. (Give your answers correct to 2 significant figures if
necessary.)
(c) What is the accumulated sales volume 2 years after the launch of the domestic game? (Give your
answer correct to 1 decimal place.)
P. 21
93. The accumulated sales volume N (in thousands) t months after the publication of a magazine can be
modelled by N  Atekt , where A and k are constants. The following table shows the accumulated sales
volume in the first 10 months of the publication of the magazine.
Time t (months)
2
4
6
8
10
Accumulated sales volume N
(thousands)
0.956
2.029
3.232
4.576
6.074
(a) Plot the graph of ln
N
against t.
t
(b) Use the graph to find the values of A and k.
(c) What is the accumulated sales volume 30 months after the publication of the magazine?
(Give your answers correct to 3 significant figures if necessary. )
P. 22
94. After the implementation of the policy prohibiting elephant hunting in a district for t years, the number of
15ebt
elephants N (in thousands) can be modelled by N  bt
(t  0) , where a and b are constants.
e a
(a) Prove that N 
15
bt
ae
1
.
15
(b) Express ln(  1) as a linear function of t.
N
(c) It is given that the slope and the intercept on the vertical axis of the graph of the linear function
obtained in (b) are 0.05 and 4.317 respectively. Find the values of a and b. (Give your answers
correct to 2 significant figures if necessary.)
(d) After implementing the policy for a long period of time, will the number of e lephants be more than
20 000? Explain briefly.
95. After taking a medicine for t hours, the concentration (in mg/ cm3) of the medicine in the bloodstream of a
patient is given by C (t )  kt ne t , where k and n are constants.
(a) Express ln[et C (t )] as a linear function of ln t .
(b) It is given that the slope and the intercept on the vertical axis of the graph of the linear function
obtained in (a) are 3.4 and 2.996 respectively. Find the values of k and n.
(c) Find the average rate of change of the concentration of medicine in the bloodstream in the first 3
hours.
(Give your answers correct to 3 significant figures if necessary. )
P. 23
96. A researcher finds that the number of bacteria N(t) (in thousands) in a test tube after t hours from the start
of the experiment for the cultivation of bacteria can be modelled by N (t )  1.1t  at n, where a and n are
constants. The following table shows the corresponding values of t and N in the test tube.
Time t (hours)
6
12
18
24
Number of bacteria N
(thousands)
2.26
3.83
6.41
10.83
(a) Express ln[N (t )  1.1t ] as a linear function of ln t .
(b) Plot the graph of ln[N (t )  1.1t ] against ln t .
(c) Use the graph to find the values of a and n.
(d) After 10 hours from the start of the experiment, how many bacteria are there in the test tube?
(Give your answers correct to 2 significant figures if necessary.)
P. 24
97. A study finds that the level L(t) of a computer game reached by a player can be modelled by
L(t )  0.35t  atent , where t (in hours) is the accumulated number of hours the player spent in playing the
computer game, and a and n are constants. The following table shows the levels reached in the first 50
hours.
Accumulated number of hours t
10
20
30
40
50
Level L(t)
18.32
28.95
34.89
38.10
39.81
(a) Express ln[
L(t )
 0.35] as a linear function of t.
t
(b) Use the graph to find the values of a and n. (Give your answers correct to 1 significant figure.)
(c) The accumulated number of hours of a player is 100, find the level reached by the player.
your answer correct to the nearest integer.)
P. 25
(Give