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CHAPTER 4 LOGARITHMIC FUNCTIONS
EXERCISE 4.1 Section 4.1 Properties and graphs of logarithmic functions (page 95)
(b) Since the points plotted in (a) lie approximately in a straight line, thus x and y are related by y k . xn =
(c) log10 k = 1.18 k = 15.1 (3 sig. fig.)
n = 118
0. 0.82
0.7−
−
= −0.514 (3 sig. fig.)
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CHAPTER 4 LOGARITHMIC FUNCTIONS
2. (a) t 5 10 15 20 25
s 120 490 1 100 1 965 3 070
log10 t 0.699 1 1.176 1.301 1.398
log10 s 2.079 2.690 3.041 3.293 3.487
(b) log10 p = 0.65 ∴ p = 4.47 (3 sig. fig.)
n = 2.690
10 0.65 0−
−.
= 2.04 (3 sig. fig.)
92
SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
3. (a) d 108 150 778 1 427 2 875 4 497
T 0.6 1.0 11.9 29.5 84.0 164.8
log10 d 2.03 2.18 2.89 3.15 3.46 3.65
log10 T −0.22 0 1.08 1.47 1.92 2.22
(b) Since the points plotted in (a) are approximately on a straight line, the data are consistent with a relation of the form T = kd . n
(c) n = 2.7 0.54 2.5
−
−
= 1.47 (3 sig. fig.)
2.7
4
log
010−
−
k =
2.7 0.54 2.5
−
− (same slope)
∴ log10 k = −3.17 k = 6.76 × 10−4 (3 sig. fig.)
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CHAPTER 4 LOGARITHMIC FUNCTIONS
4. (a) d 5 12 15 20 28
F 7.2 1.3 0.8 0.5 0.2
log10 d 0.70 1.08 1.18 1.30 1.45
log10 F 0.86 0.11 −0.10 −0.30 −0.70
From the graph of log10 F against log10 d, we see that the points plotted are approximately on a straight line. It reveals that the data support the claim.
(b) log10 k = 2.3 ∴ k = 200 (3 sig. fig.)
n = 2.3 −
−
1.50 0.4
= −2
(c) From the result of (b), F = 200d − 2
When d = 3, F = 200(3−2) = 22.2
94
SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
5. (a) x 6 15 22 30 43
y 4.0 2.6 2.1 1.8 1.6
ln x 1.79 2.71 3.09 3.40 3.76
ln y 1.39 0.96 0.74 0.59 0.47
(b) Let y = kx n . Then ln k = 2.25 k = 9.49 (3 sig. fig.)
and n = 2.25
0−
−
0.8 3
= −0.483 (3 sig. fig.) ∴ y = 9.49 0 483x − .
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CHAPTER 4 LOGARITHMIC FUNCTIONS
6. (a) t 4 8 12 16 20 24
S 55 157 290 450 620 830
ln t 1.39 2.08 2.48 2.77 3.00 3.18
ln S 4.01 5.06 5.67 6.11 6.43 6.72
(b) Let S = kt . n
Then ln k = 1.9 k = 6.69 (3 sig. fig.)
and n = 5 2. 1.92.2 0
−
−
= 1.5 ∴ S = 6 6 9 1. .5t (c) When t = 10, S = 6.69 (101.5) = 212
96
SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
7. (a) P = a ekt
∴ ln P = ln (a ekt) = ln a + ln ekt
= kt + ln a (b) t 1 2 3 4 5 P 160 276 441 743 1 210 ln P 5.075 5.620 6.089 6.611 7.098
(c) ln a = y-intercept of the line in (b) = 4.6 ∴ a = e4.6
= 99.5 (3 sig. fig.) k = slope of the line in (b)
= 05
6.41.7−−
= 0.5
(d) From the result of (c), P= 99.5 e0.5t
When t = 7, P = 99.5 e0.5(7)
= 3 294.99 i.e. The population of the bacteria is approximately 3 295 after 7 hours.
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CHAPTER 4 LOGARITHMIC FUNCTIONS
8. (a) H(t) = h ebt
∴ ln [H(t)] = ln (h ebt) = ln h + ln ebt
= bt + ln h (b) t 1 2 3 4 5
H(t) 2.9 3.3 4.1 4.5 5.2
ln [H(t)] 1.065 1.194 1.411 1.504 1.649
(c) b = slope of the line in (b)
= 04
9.05.1−−
= 0.15 ln h = y-intercept of the line in (b) = 0.9 ∴ h = e0.9
= 2.46 (3 sig. fig.)
(d) From the result of (c), H(t) = 2.46 e0.15t
When H(t) = 6.5, 6.5 = 2.46 e0.15t
∴ 0.15t = ln 46.25.6
t = 6.478 i.e. The tree will be 6.5 m tall after 6.5 years.
(e) There is a constraint because the tree will not grow at such a rapid rate after 10 years. Besides, a tree has a limiting height and it cannot grow at this rate forever.
98
SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
9. (a) t 0 2 4 6 8 10 M 8.0 7.5 7.0 6.7 6.3 5.8 ln M 2.079 2.015 1.946 1.902 1.841 1.758
(b) M = a e–kt
∴ ln M = ln (a e–kt) = ln a + ln e–kt
= ln a − kt ln a = y-intercept of the line in (a) = 2.062 5 ∴ a = 7.87 (3 sig. fig.) −k = slope of the line in (a)
= 210275.1
−−
= −0.031 3 ∴ k = 0.031 3 (3 sig. fig.)
(c) From the result of (b), M = 7.87 e–0.031 3t
When t= 15, M= 7.87 e–0.031 3(15)
= 4.921 2 i.e. The mass of the substance will be 4.92 grams.
(d) 287.7 = 7.87 e–0.031 3t
∴ −0.031 3t = ln 21
t = 22.145 i.e. The half-life of the substance is 22.1 months.
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CHAPTER 4 LOGARITHMIC FUNCTIONS
10. (a) t 30 60 90 120 150
Q 1 776 1 150 723 450 290
ln Q 7.482 7.048 6.583 6.109 5.670
If Q = a ebt, then ln Q = ln (a ebt) = ln a + ln ebt
= bt + ln a Since the plotted points of ln Q against t are approximately on a straight line, the above
relation is valid.
(b) ln a = y-intercept of the line in (a) = 7.9 ∴ a = 2 697 (to the nearest integer) b = slope of the line in (a)
= 01509.77.5
−−
= −0.015 (2 sig. fig.)
(c) From the result of (b), Q = 2 697 e–0.015t
When t = 0, Q = 2 697 e–0.015(0)
= 2 697 i.e. The battery can hold 2 697 C of charge.
(d) 100 = 2 697 e–0.015t
∴ −0.015t = ln 697 2
100
t = 219.65 i.e. The battery can last for 220 minutes.
100
SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
11. (a) x 2 4 6 8 10 12
y 3.5 3.8 4.1 4.3 4.5 4.6
e y 33.1 44.7 60.3 73.7 90.0 99.5
(b) y = ln (a + bx) ∴ = a + bx e y
Hence, a = y-intercept of the line in (a) = 20 b = slope of the line in (a)
= 68 20
7 0−
−
= 6.86 (3 sig. fig.)
12. (a) x 1 3 5 7 9
y 3 34 85 160 255
yx2 3 3.78 3.4 3.27 3.15
log102
xx
0 0.053 0.028 0.017 0.012
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CHAPTER 4 LOGARITHMIC FUNCTIONS
(b) y = ax2 + b log10 x
∴ y
x 2 = 2
10log + x
xba ⋅
Hence, a = y-intercept of the line in (a) = 3 b = slope of the line in (a)
= 3 3 . −
−
30.02 0
= 15
13. (a) ln ⎥⎦
⎤⎢⎣
⎡−1
)(500
tP = ln ⎥
⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛+
÷−
1 1
500500btea
= ln (1 + a e–bt − 1) = ln (a e–bt) = ln a + ln e−bt
= ln a − bt
(b) Draw the graph of ln ⎥⎦
⎤⎢⎣
⎡− 1
)(500
tPagainst t.
t 1 2 3 4 5 P(t) 120 142 170 209 236
ln ⎥⎦
⎤⎢⎣
⎡−1
)(500
tP 1.153 0.925 0.663 0.331 0.112
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SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
ln a = y-intercept of the line in the figure = 1.475 ∴ a = 4.37 (3 sig. fig.) −b = slope of the line in the figure
= 15
2.11.0−−
= −0.275 ∴ b = 0.275
(c) From the result of (b),
P(t) = te 275.0 37.41
500−+
(i) When t = 0,
P(t)= 0 275.0 37.41
500×−+ e
= 93.11 i.e. The population of frogs is 93.
(ii) When t = 12,
P(t) = 12 275.0 37.41500
×−+ e
= 430.60 i.e. The population of frogs is 431. (d) When t tends to infinity, 4.37 e–0.275t tends to zero.
∴ P(t) = 01
500+
= 500 i.e. The population of frogs in the pond will eventually be 500. 14. (a) y = 50 + 30 (1 − p e–qx)
30
50−y = 1 − p e–qx
1 − 30
50−y = p e–qx
∴ ln ⎟⎠⎞
⎜⎝⎛ −−
30501 y = ln (p e–qx)
= ln p + ln e–qx
= ln p − qx
Hence, ln ⎟⎠⎞
⎜⎝⎛ −−
30501 y and x are linearly related.
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CHAPTER 4 LOGARITHMIC FUNCTIONS
(b) Draw the graph of ln ⎟⎠⎞
⎜⎝⎛ −−
30501 y against x.
x 1 2 3 4 5 y 55 62 64 68 69
ln ⎟⎠⎞
⎜⎝⎛ −−
30501 y −0.182 −0.511 −0.629 −0.916 −1.003
ln p = y-intercept of the line above = 0 ∴ p = 1 −q = slope of the line above
= 03
063.0−−−
= −0.21 ∴ q = 0.21
(c) From the result of (b), y = 50 + 30(1 − e–0.21x) 75 = 50 + 30(1 − e–0.21x)
e–0.21x = 61
∴ −0.21x = ln 61
x = 8.532 i.e. 8.5 kg per m2 of fertilizer should be applied.
(d) (i) When x = 0, y = 50 + 30[1 − e–0.21(0)] = 50 i.e. The yield will be 50 kg per m2. (ii) When x is very large, e–0.21x ≈ 0 ∴ y = 50 + 30(1 − 0) = 80 i.e. The yield will be 80 kg per m2.
104
SECTION 4.5 LOGARITHMIC TRANSFORMATIONS
15. (a) N = atbe–t
ln N = ln (atbe–t) = ln a + ln tb + ln e–t
= ln a + b ln t − t ∴ ln N + t = ln a + b ln t Hence, ln N + t and ln t are linearly related.
(b) Draw the graph of ln N + t against ln t. t 1 2 3 4 5
N 55 160 200 172 128
ln t 0 0.693 1.099 1.386 1.609
ln N + t 5.007 7.075 8.298 9.147 9.852
ln a = y-intercept of the line above = 5 ∴ a = 148.4 (1 d.p.) b = slope of the line above
t = 6.622 i.e. The amount of polluted material will become 300 after 7 months.
9. (a) (i) When S = 21 S0,
21 S0 = S0 e–0.12t
∴ −0.12t = ln 21
t = 5.776 i.e. The sales will reduce to half after 5.8 months.
110
REVISION EXERCISE 4
(ii) When S = 30% of S0, 0.3S0 = S0 e–0.12t
0.3 = e–0.12t
∴ −0.12t = ln 0.3 t = 10.033 i.e. The sales will diminish to 30% of the original amount after 10.0 months. (b) A = the sales at the beginning of the campaign = S0 e–0.12 × 6
= S0 e–0.72
When S = S0, S0 = S0 e–0.72 × e0.08t
1 = e–0.72 + 0.08t
∴ −0.72 + 0.08t = 0 t = 9 i.e. The sales will resume to S0 after 9 months. 10. (a) 2 log2 x = y + 1 . . . . . . (1) log2 (2x) = y + 5 . . . . . . (2) (1) − (2), 2 log2 x − log2 (2x) = −4 ∴ log2 x2 − log2 (2x) = −4
log2
2
2x
x = −4
x2
= 2−4
x = 18
. . . . . . . . (3)
Putting (3) into (1), 2 log2 18
= y + 1
2 (−3) = y + 1 ∴ y = −7
(b) + = 4 e x2 xe 23 −
− 4 + = 0 e x2 xe 23 −
(ex − ) (e − ) = 0 xe− x xe−3 ∴ = or ex xe− xe−3 = 1 or 3 e x2
2x = 0 or ln 3
x = 0 or 12
ln 3
(c) log10 (9 − 1) = logx10 (3 + 1) + 1 x
1 + 31 9log10 x
x − = 1
1 + 3
1) + (3 1) 3(log10 x
xx − = 1
log10 (3 − 1) = 1 x
− 1 = 10 3x
= 11 3x
∴ x = loglog
10
10
113
= 2.183 (4 sig. fig.)
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CHAPTER 4 LOGARITHMIC FUNCTIONS
11. (a) (ln x)2 = 2 ln x (ln x)2 − 2 ln x = 0 ln x (ln x − 2) = 0 ln x = 0 or 2 ∴ x = 1 or e2
(b)
(c) From the graph in (b), (ln x)2 > 2 ln x when x < 1 or x > e2
(d) (i) Substitute the point (e2, 5) into y = 2 ln x + k, 5 = 2 ln e2 + k 5 = 2(2) + k ∴ k = 1 (ii)
Note: The graph of y = 2 ln x + 1 can be obtained by shifting the graph of y = 2 ln x
in (b) 1 unit upward.
112
REVISION EXERCISE 4
12. (a) m = 5 e−0.4t
When t = 0, m = 5 e−0.4(0)
= 5
(b) If the mass becomes 54
g,
54
= 5 e−0.4t
−0.4t = ln 14
t = ln40 4.
= 3.466 i.e. The required time is 3.466 hours.
(c) Let t1 and t2 be the times when the mass becomes 3 g and 2.9 g respectively. 3 = . . . . . . . . . . . . (1) 14.0 5 te−
2.9 = . . . . . . . . . . . . (2) 24.0 5 te−
From (1), 35
= 14.0 te−
−0.4t1 = ln 35
t1 = 53 ln
4.01
−
Similarly, from (2), t2 = 59.2 ln
4.01
−
t2 − t1 = ⎟⎠⎞
⎜⎝⎛ −
− 53 ln
59.2 ln
4.01
= 39.2 ln
4.01
−
= 0.084 75 i.e. The required time = 0.084 75 hour
(d) (i) m = 5 e–0.4t
ln m = ln (5 e–0.4t) = ln 5 + ln e–0.4t
= ln 5 − 0.4t ∴ ln m and t are connected by the equation in the form ln m = a + bt where a and b
are constants. Hence, the graph of ln m against t is a straight line. (ii) The slope is −0.4. The intercept on the ln m axis is the value of ln m when t = 0. i.e. ln m = ln 5 ∴ The intercept = ln 5
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CHAPTER 4 LOGARITHMIC FUNCTIONS
13. (a) (i) The half-life is the time at which P = 50. 50 = 100 e–0.000 12t
∴ −0.000 12t = ln 10050
t = 5 780 (3 sig. fig.) i.e. The half-life of carbon-14 is 5 780 years. (ii) When P = 30,
30 = 100 e−0.000 12t
∴ −0.000 12t = ln 10030
t = 10 033.11 i.e. The bone has been buried for 10 033 years.
(b) (i) Suppose the man has died for t0 hours at 11 p.m. Now T0 = 16. When t = t0, T = 25. ∴ 25 − 16 = (37.5 − 16) 0kte−
∴ t0 = 1.074 ∴ The man has died for 1 hour at 11 p.m. i.e. The murder was committed at 10 p.m. (iii) No, carbon-dating cannot be employed to estimate the time of death as only a very little
amount of carbon-14 would have decayed. The change in percentage of carbon-14 cannot be measured accurately enough to evaluate the murder time.
114
REVISION EXERCISE 4
14. (a) I = kV n
log10 I = log10 k + n log10 V Thus, the graph of log10 I against log10 V is a straight line.
V 160 180 200 220 240
I 52 70 93 121 152
log10 V 2.204 2.255 2.301 2.342 2.380
log10 I 1.716 1.845 1.968 2.083 2.182
n = slope of the graph
= 275.2 375.29.1 175.2
−−
= 2.75 log10 k = y-intercept of the graph
∴ 0 275.2
log 9.1 10
−− k
= 2.75
log10 k = −4.356 k = 4.4 × 10−5
(b) Thus, I = 4.4 × 10−5 V 2.75. When V = 230, I = 4.4 × 10−5 (2302.75) = 137
(c) If V increases to V1 = 2V, then I becomes I1 = 4.4 × 10−5 (2V)2.75
= 6.727 (4.4 × 10−5 V 2.75) ∴ The percentage increase in I = (6.727 − 1) × 100% = 572.7%
(c) From the straight line graph in (b)(ii), we can say that ln Q = ln a + bt where a and b are constants. Hence, Q = a ebt. ln a = y-intercept of the line in (b)(ii) = 3.9 ∴ a = 49.4 (3 sig. fig.) b = slope of the line in (b)(ii)
= 05
9.39.8−−
= 1 ∴ The equation connecting Q and t is Q = 49.4 et.
(d) When t = 7, Q = 49.4 e7
= 54 200 (3 sig. fig.) i.e. The amount of algae will be 54 200 after 7 days. 16. (a) (i) ln [P(t) + 5] = ln [a ebt − 5 + 5] = ln (a ebt) = ln a + ln ebt
(iii) ln a = y-intercept of the line in (ii) = 2.3 ∴ a = 10 (2 sig. fig.) b = slope of the line in (ii)
= 08
3.25.3−−
= 0.15 (iv) Hence, the equation connecting P and t is P(t) = 10 e0.15t − 5. When t = 10, P(10) = 10 e0.15(10) − 5 = 40 i.e. The amount of product is 40 grams when t = 10.