Chapter 7 Exponential Functions 7A p.2 Chapter 8

1

Chapter 7 Exponential Functions

7A p.2

7B p.16

Chapter 8 Logarithmic Functions

8A p.27

8B p.39

8C p.48

8D p.56

Chapter 9 Rational Functions

9A p.64

9B p.75

9C p.86

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2

F4B: Chapter 7A

Date Task Progress

Lesson Worksheet

○ Complete and Checked ○ Problems encountered ○ Skipped

(Full Solution)

Book Example 1

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(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 7A Level 1


Teacher’s Signature

___________ ( )

Maths Corner Exercise ○ Complete and Checked Teacher’s

3

7A Level 2 ○ Problems encountered ○ Skipped

Signature ___________ ( )

Maths Corner Exercise 7A Multiple Choice

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E-Class Multiple Choice Self-Test

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4

4B Lesson Worksheet 7.0 (Refer to Book 4B P.7.3)

Objective: To review positive integral indices, zero index, negative integral indices and laws of integral

indices.

Positive Integral Indices

Without using a calculator, find the value of each of the following expressions. [Nos. 1–6]

1. 15 = 2. 33 = �Review Ex: 1, 2

3. (−4)3 = 4. (−9)2 =

5.

4

2

1

=

) () (

1=

6.

3

10

1

− =

) () (

1=

Zero Index and Negative Integral Indices


7. (−12)0 = 8.

0

4

3

= �Review Ex: 3–6

9. 2−3 =) () (

1 10. (−4)−2 =

) () (

1

=

=

Laws of Integral Indices


11. 37 ⋅ 3–4 12. 4

7

5

5

= ( )( ) + ( ) = ( )( ) − ( )

= =

= =

13. (23)2 14. 43

3

2

1

−

= 2( ) × ( )

=

=

n

m

a

a= am – n am ⋅ an = am + n

(ab)n = anbn (am) n = amn

�Review Ex: 7–18

=

5

15.

2

8

7

16. 2–3 × 2–1

=) (

) (

) (

) (

=

=

17. (90 × 3–1)–2 18.

2

25

3−

×

31

5

2

−

=

=

Simplify the following expressions and express the answers with positive indices. [Nos. 19–22]

19. a9 ⋅ a–6 20. 3

5

a

a �Review Ex: 19–27

= =

21. 4

32 )(−

b

ba 22. (ab)–3

3

b

a

= ) (

) () (

b

ba =

=

��Level Up Question��

23. Is the result of 52 014 × (–0.2)2 015 less than 0? Explain your answer.

n

b

a

=

n

n

b

a

Is the value of (–0.2)2 015 equal to 0?

6

4B Lesson Worksheet 7.1A & B (Refer to Book 4B P.7.4)

Objective: To understand the root of a number and rational indices.

Review: Positive Integral Indices, Zero Index and Negative Integral Indices


1. (−5)2 = 2. 110 = 3. 4−3 =) () (

1

=

The Root of a Number

If xn = a, where n is a positive integer, then x is an nth root of a.


4. 49 5. 4 16 � Ex 7A: 1

= =

6. 3 27− 7. 3 64−

= =

Rational Indices

If a > 0, m and n are integers, and n > 0, then

(i) na

1

= n a (ii) n

m

a = mn a )( =n m

a

Express each of the following expressions in the form ap, where p is a rational number and a > 0. [Nos. 8–11]

8. 6 a 9. 3)( a � Ex 7A: 2

= ) (

1

a = ) (

) (

a

10. 5 7

a 11. 23 )( −a

= =

Instant Example 1 Instant Practice 1

Without using a calculator, find the values of 2

1

16

and 3

4

8 .

2

1

16 = 16 3

4

8 = 43 )8(

= 4 = 24

= 16

Without using a calculator, find the values of 3

1

27

and 3

2

64 .

3

1

27 = 3 3

2

64 = 23 ) (

= = ( )2

=

�

49 denotes the positive

square root of 49 only.

( )3 = −27

( )4 = 16

7


12. 2

1

49 = 13. 3

1

125 = � Ex 7A: 3, 4

=

14. 2

3

9 = 3) ( 15. 4

5

16 =

= ( )3

=


Without using a calculator, evaluate

3

2

27

1−

.

3

2

27

1−

=

2

3

27

1−

=

2

3

1−

=2

3

1

1

= 9

Without using a calculator, evaluate

2

3

25

1−

.

2

3

25

1−

=

3

) (

1−

=

3

) (

1−

=3

) (

1

1

=


16. 2

3

16

1−

=

3

) (

1−

17.

5

2

32

1−

= � Ex 7A: 5, 6

=

3

) (

1−

=


18. Without using a calculator, find the value of each of the following expressions.

(a) 3

125

8 (b)

4

1

81

16−

8

4B Lesson Worksheet 7.1C (Refer to Book 4B P.7.9)

Objective: To understand the laws of rational indices.

Review: Laws of Integral Indices


1. 4

35

a

aa−⋅

= a( ) + ( ) − ( ) 2.

214

−

c

ba=

2

2) (

) (

)(a

= a( ) =) () (

) () (

cb

a ×

=

=

Laws of Rational Indices

Let a > 0, b > 0, p and q be any rational numbers.

(i) ap ⋅ aq = ap + q (ii) q

p

a

a= ap − q (iii) (ap)q = apq

(iv) (ab)p = apbp (v) p

b

a

=

p

p

b

a Note: a–q =

qa

1


3. 3

1

2 × 3

5

2 = 2( ) + ( ) 4. 2

7

3 ÷ 2

5

3 = 3( ) − ( )

= 2( ) = 3( )

= =

5.

2

1

4

3

2

3

1616

16

×

= 16( ) − ( ) − ( ) 6. 3

2

4

3

)9( = 9( ) × ( )

= 16( ) = 9( )

= =

7. 3

136 )72( × = 2( ) × ( ) × 7( ) × ( ) 8.

2

1

2

4

5

2

=

) () (

) () (

5

2×

×

= =)(

2 ) (

=

�

am ⋅ bn = am + n

n

m

a

a= am – n

a−n =n

a

1

(am) n = amn

(ab)n = an bn

n

b

a

=

n

n

b

a

9


9. a × 4

1

a = a( ) + ( ) 10. 2a × 4

1−

a = a( ) + ( ) 11. 2

1

a ÷ 3

1

a = a( ) − ( ) � Ex 7A: 11–16

= = =

12.

5

1

b

b= b( ) − ( ) 13. 3

1

4

9

)(a = a( ) × ( ) 14. (a4b)2 = a( ) × ( )b( )

= = =


Simplify 5

1

a ÷ 4

1

5

2

)(a and express the answer with

positive index.

5

1

a ÷ 4

1

5

2

)(a = 5

1

a ÷ 4

1

5

2×

a

= 5

1

a ÷ 10

1

a

= 10

1

5

1−

a

= 10

1

a

Simplify 3

1

4 )(a ÷3 2

a and express the answer with

positive index.

3

1

4 )(a ÷3 2

a = a( ) × ( ) ÷ a( )

= a( ) ÷ a( )

= a( ) − ( )

= Simplify the following expressions and express the answers with positive indices. [Nos. 15–18]

15. 3

2

9 )(ab = a( )b( ) × ( ) 16. 6

1

3

2

b

a=

) () (

) () (

×

×

b

a � Ex 7A: 17–25

= =

17. 2

1

4

186 )( aba ÷ = 18. 5

a ×2

1

a

b=


19. Let n be a rational number.

(a) Express 92n as a power of 3. (b) Hence, simplify n

nn

2

1

9

33 +⋅.

10

New Century Mathematics (Second Edition) 4B

7 Exponential Functions

� Consolidation Exercise 7A

Level 1

1. Without using a calculator, find the value of each of the following expressions.

(a) 3 27 (b) 4 256 (c) 6 64

(d) 3 8− (e) 3 125− (f) 5 243−

2. Express each of the following expressions in the form xk, where k is a rational number and x > 0.

(a) 3 x (b) 4 x (c) 5 3x

(d) 4 5−x (e) 73 )( x (f) 56 )( −

x


3. 3

1

64 4. 2

3

49 5. 5

3

32

6. 3

7

8−

7. 3

1

125−

8. 3

4

27

1

9. 2

3

9

1−

10.

2

1

25

36

11.

4

3

81

16−

Use a calculator to find the value of each of the following expressions. [Nos. 12–17]

(Give the answers correct to 3 significant figures.)

12. 5 16 13. 54 )32( 14. 29 )121( −

15. 6

5

80 16. 5

8

25

12

17.

3

1

5

6−

Simplify each of the following expressions and express the answers with positive indices. [Nos. 18–31]

18. (a) 3

1

2aa × (b) 3

2

2

3−

×bb

(c) 3

4

5

2

cc ×−

(d) 6

1

2

1−−

× dd

11

19. (a)

2

1

4

1

x

x (b)

1

3

2

−y

y

(c)

3

1

2

1

p

p−

(d)

4

3

3

2

−

−

q

q

20. (a) 2

1

4 )(h (b) 24

3

)( −k

(c) 6

5

10

3

)(−

m (d) 3

4

8

3

)(−−

n

21. (a) 3

4

3)( −m (b)

63

2

)(

1

−n

(c) 7

3

7

1

c (d)

9

9

4

1

−

−

k

22. (a) a × 32

1

)(a (b) 23

4

3

2

)( −× xx (c) 4

1

22

5

)( −−

× yy

23. (a) a2 ÷ 3

1

4 )(a (b) 2

1

32

3

)( mm ÷− (c) 6

5

43

2

)(−

÷ nn

24. (a) 2

1

a × 3 a (b) 5 x × 34 )( x (c) 3 y × 65 )( −y

25. (a) 4 a ÷ a (b) 3 4

m ÷4 3

m (c) 53 )( −n ÷ 3−

n

26. (a) 5

2

4

55 )( ba (b) 4

3

3

8

)(−

xy (c) 6

1

2

3

)(−−

sr

27. (a) 6

1

2 )( ba × 3

1

a (b) 3

1

34 )( yx × y−2 (c) 2

3

24 )(−

−kh × 04

kh

28. (a) 3

4

23 )( dc− ÷ 3

2

d (b) 63

4

3

1

)(−

ba ÷ (ab−2) (c) 24

1

2

3

)( −−

sr ÷ )( 2

3

1sr

−

29. (a) 3

2

3

2

b

a×

b

1 (b)

3

1

y×

4

3

4

−

y

x

30. (a) 2

1

4

b

a÷

b

a 4

3

(b)

5

2

2

1

−

m

n÷

5

1

5

3−

n

m

31. (a) 2

1

3

2

)4( ba (b) 2

1

2

1

2 )9(−

−ba

12

Simplify each of the following expressions, where n is a rational number and n ≠ 0. [Nos. 32–35]

32. nnaa

2

)( × 33.

n

n

a

a

3

1

34. na )4( 2

1−

× 3

1

)( na

− 35.

3

4

32

)27(

)(

n

n

a

a

−

Level 2


36. 4

81

256 37. 3

343

512−

38. 2

1

25

121−

39.

3

2

250

432−

Simplify each of the following expressions and express the answers with positive indices. [Nos. 40–53]

40. (a) 2

1

3)25(−

a × 3

1

a (b) 3 28x × 4

3−

x (c) 4 316 −y × 3 4

y

41. (a) 53

1

4

)(

36

c

c

−

(b)

3

1

2

35

)125(

)(

−

m

m (c)

3

3 27

−

−

p

p

42. (a) 2

1

a × 3 a ×4 1−

a (b) 3 2

k ÷ 3)( −k × k

(c) 3

1

n ×3 2

n ÷ 32

1

)3( −−

n (d) 5h ÷ 2

1

2 )4( h ÷ 43 )( h

43. (a) 2

3

3 2

a

a× a (b)

3

3

1

−

−

b

b× 2

3

b

44. (a)

24 3

x

x÷ 3 x (b)

3

2

1

2

−y

y÷ 2

1−

y

45. (a) 43

3 22

ba

ba ⋅ (b)

yx

yx22

3

4

1

)(−

46. (a) 2

b

a÷

3 5

3

1

b

a−

(b)

4

3

4 2

y

x÷

3 1

4

1

−x

y

13

47. (a) 23 123

1

)( −baa (b) 3232

3

2 )( yxyx−

−

48. (a) 323

12

kh

kh−

−

(b)

2

3 2

3

2

ab

ba

49. (a) 33

1

2 )( ba− × 2

ab (b) 3 42 −yx × 3

2

2 )(−

yx

50. (a) 3

1

2

3

134 )(4 nmnm−× (b) 13 −

qp × 3 28 qp−

51. (a) 3

1

2

3

b

a× 3

2a

b (b)

5

39

n

m−

×2

1

2

3−

m

n

52. (a) 4

3

2

8

−b

a÷

3 2

2

1

a

b (b)

2

1

3

2

m

mn÷ 3

2

8n

m−

53. (a) xx (b) aaa

(c) ba 814 (d) 53 1625 st

Simplify each of the following expressions, where all the unknowns are rational numbers. [Nos. 54–67]

54. k

k

4

2 13 +

55. 22

1

3

27+

−

x

x

56. 146

366+

⋅a

aa

57. nn

n

525

55 2

⋅

⋅−

58. 46

12

2

16+

−

x

x

59. 32

12

27

9n

n

−

−

60. 3a + 2⋅3a + 3a + 1 61. 42k + 42k + 1 + 42k + 2

62. 1

1

22

2−

+

+ nn

n

63. k

kk

2

1212

3

33 −+ −

64. 1313

1323

44

44−+

−+

−

+aa

aa

65. 21

11

66

66−+

+−

−

−nn

nn

66. 2

2

)3(2

9)3(3x

xx + 67.

131

3

28

)2(3++ − nn

n

14

Answers

Consolidation Exercise 7A

1. (a) 3 (b) 4

(c) 2 (d) −2

(e) −5 (f) −3

2. (a) 3

1

x (b) 4

1

x

(c) 5

3

x (d) 4

5−

x

(e) 3

7

x (f) 6

5−

x

3. 4 4. 343

5. 8 6. 128

1

7. 5

1 8.

81

1

9. 27 10. 5

6

11. 8

27 12. 1.74

13. 76.1 14. 0.344

15. 38.5 16. 0.309

17. 0.941

18. (a) 3

7

a (b) 6

5

b

(c) 15

14

c (d)

3

2

1

d

19. (a)

4

1

1

x

(b) 3

5

y

(c)

6

5

1

p

(d) 12

1

q

20. (a) h2 (b)

2

3

1

k

(c)

4

1

1

m

(d) 2

1

n

21. (a) 4

1

m (b) n4

(c) 3

1

c (d)

4

1

k

22. (a) 2

5

a (b) 2

1

x

(c) 3

1

y

23. (a) 3

2

a (b) 5

1

m (c) n4

24. (a) 6

5

a (b) 20

19

x (c)

15

13

1

y

25. (a)

4

3

1

a

(b) 12

7

m (c)

6

1

1

n

26. (a) 2

1

2ba (b)

2

4

3

y

x (c)

6

1

4

1

s

r

27. (a) 6

1

3

2

ba (b) y

x 3

4

(c) 2

3

h

k

28. (a) 4

2

c

d (b)

6b

a (c)

2

4

s

r

29. (a) 3

3

4

b

a (b)

4

3

1

x

30. (a)

ba 4

1

1 (b)

2

1

n

m

31. (a) 2

1

3

1

2 ba (b)

4

1

3b

a

32. 2

1+

na 33. 31

n

a−

34.

6

5

4n

n

a

35. 81

2na

36. 3

4 37.

7

8−

38. 11

5 39.

36

25

40. (a)

6

7

5

1

a

(b)

12

1

2

x

(c) 12

7

2y

15

41. (a) 3

1

6c− (b) 15

19

5m (c) 6

11

3p−

42. (a) 12

7

a (b) 6

19

k

(c)

2

1

27

n

(d) 6

1

2

1h

43. (a) a (b) b

1

44. (a)

6

5

1

x

(b) 2

3

y

45. (a)

3

4

2

1

b

a (b)

2

7

2

1

1

yx

46. (a)

3

1

6

5

b

a (b)

y

x 6

1

47. (a)

3

2

3

5

b

a (b)

2

5

2

3

x

y

48. (a) k

h3

5

(b) 3

2

3

2

ba

49. (a)

2

11

2

a

b (b)

23

2

1

yx

50. (a) 23

5

2 nm (b)

6

1

6

5

2

q

p

51. (a)

3

1

3

1

b

a (b)

42

1

3

nm

52. (a) ba 3

11

(b) 3

4

2mn

53. (a) 4

3

x (b) 4

7

a

(c) 4

1

2

1

6 ba (d) 4

5

2

3

10 st

54. 2k + 1 55. 31 − 5x

56. 6−a − 1 57. 53n + 1

58. 2x − 4 59. 3

87

3

−n

60. 6⋅3a 61. 21⋅42k

62. 3

4 63.

3

8

64. 3

13 65.

43

42−

66. 2 67. 2

1

16

F4B: Chapter 7B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)



(Full Solution)

Maths Corner Exercise 7B Level 1



___________ ( )




___________ ( )

Maths Corner Exercise 7B Multiple Choice


___________ ( )



_________

17

4B Lesson Worksheet 7.2A (Refer to Book 4B P.7.15)

Objective: To understand the properties of exponential functions and their graphs.

Exponential Functions

Let x and y be variables and a be a constant, where a > 0 and a ≠ 1.

Then y = ax is called an exponential function with base a.

Properties of Graphs of the Exponential Functions y = ax

Range of a a > 1 0 < a < 1

Graph of y = ax

Common

properties

1. The graph cuts the y-axis at the point (0 , 1). 2. The graph does not cut the x-axis. It lies above the x-axis. For each value

of x, the value of y is greater than 0.

3. The graphs of

x

ay

=

1and y = ax are the images of each other when

reflected in the y-axis.

Different

properties

1. The graph slopes upward from left to right.

2. When the value of x increases, the value of y increases and so does the rate of increase of y.

1. The graph slopes downward from left to right.

2. When the value of x increases, the value of y decreases and so does the rate of decrease of y.

1. 2.

Coordinates of P = ( , ) Coordinates of P = ( , )

When the value of x increases, When the value of x increases,

the value of y ( increases / decreases ). the value of y ( increases / decreases ).

3. 4.




x

y

O

y = 8x

P

x

y

O

P

y = 0.3x

x

y

O

P

y =

x

5

1

x

y

O

y = 4x

P

x

y

0

1

y = ax

x

y

1

0

y = ax

�

4 ( < / > ) 1 0 <

5

1< 1

8 ( < / > ) 1

18

5. The figure shows the graph of y = 2x. 6. The figure shows the graph of

x

y

=

6

1.

Sketch the graph of

x

y

=

2

1in the same figure. Sketch the graph of y = 6x in the same figure.

7. The two curves in the figure represent the graphs of the exponential

functions y = 5x and y = 0.2x. Write down the corresponding

exponential function for each of C1 and C2. �Ex 7B: 3–5

C1:

C2:


functions y = 3x and y = 9x. Write down the corresponding exponential

function for each of C1 and C2.

C1:

C2:


functions y = 0.4x and y = 0.7x. Write down the corresponding

exponential function for each of C1 and C2.

C1:

C2:


10. The figure shows the graphs of the exponential functions y = 8x

and y = ax, where a is an integer. Paul claims that the value of a

can be 10. Do you agree? Explain your answer.

x

y

O

y =

x

6

1

x

y

O

y = 2x

x

y

O

C1 C2

x

y

O

C1

C2

x

y

O

C1 C2

x

y

O

y = 8x

y = ax

x

y

O

?

?

19

y

x

C2 C1

O

y

x

C3 C4

O


7 Exponential Functions

� Consolidation Exercise 7B

Level 1

1. In each of the following, when the value of x increases, does the value of the exponential function

increase or decrease?

(a) y = 6x (b) y = 0.4x (c) y =

x

6

1

(d) y = 1.2x (e) y = 3−x (f) y =x5

1

2. In each of the following, the graph of the exponential function is reflected in the y-axis. Write down

the corresponding function of each graph obtained.

(a) y =

x

4

3 (b) y = 4x (c) y = 8−x

(d) y =x9

1 (e) y = 0.3x (f) y = 2.5x


functions y = 3x and y = 7x. Write down the corresponding exponential



functions y =

x

2

1and y = 8x. Write down the corresponding


20

y

x

C5

O

C6

y

x

C8

O

C9

C7


functions y = 0.3x and y =

x

5

3. Write down the corresponding


6. The three curves in the figure represent the graphs of the exponential

functions y = 3.5x, y = 3−x and y = 0.7x. Write down the corresponding

exponential function for each of C7, C8 and C9.

7. (a) Sketch the graph of y = 2.5x in the figure.

(b) Write down the y-intercept of the graph.

(c) According to each of the following conditions, write down the

range of values of y of the function y = 2.5x.

(i) x ≥ 0

(ii) x < 0

8. (a) Sketch the graph of y = 0.6x in the figure.

(b) The graph cuts the y-axis at the point P. Write down the

coordinates of the point P.

(c) According to each of the following conditions, write down the

range of values of y of the function y = 0.6x.

(i) x ≤ 0

(ii) x > 0

7

6

5

4

3

2

1

x −2 −1 0 1 2

y

3

2

1

x −2 −1

y

0 1 2

21

9. Let y = 2−x.

(a) Complete the table below.

x −2 −1 0 1 2

y

(b) Sketch the graph of y = 2−x in the figure.

(c) Write down the coordinates of the point where the graph cuts the

y-axis.

(d) According to each of the following conditions, write down the

range of values of x of the function y = 2−x.

(i) 0 < y < 1

(ii) y ≥ 1

10. The value $P of a machine after t years can be represented by the formula P = 320 000(0.85)t. Find

the value of the machine after 4 years.

11. The selling price $V of a tablet computer after x months is given by the formula V = 2 500(0.95)x.

(a) Find the present selling price of the tablet computer.

(b) Find the selling price of the tablet computer after 6 months.

(Give the answer correct to 3 significant figures.)

12. The diameter d cm of the trunk of a tree t years after it was planted for 10 years can be represented by

the formula d = 20(1.02)t.

(a) Find the diameter of the trunk of the tree after it was planted for 10 years.

(b) Find the diameter of the trunk of the tree after it was planted for 13 years.


13. The number R of birds in a forest after t weeks can be estimated by the formula

R = 6 000 + 1 250(0.84)t. Find the number of birds in the forest after 2 weeks.

14. An experiment is conducted to study the effect of a certain drug on the growth of bacteria. t hours

after the drug is applied, the number N of bacteria remained can be represented by the following

formula:

N = k(0.9)t, where k is a constant.

It is known that the number of bacteria after 2 hours is 4 050.

(a) Find the value of k.

(b) Find the number of bacteria at the beginning of the experiment.

(c) Find the number of bacteria after 5 hours.

(Give the answer correct to the nearest integer.)

4

3

2

1

x −2 −1

y

0 1 2

22

y

x O

y = 6x

y

x O

C2

C1

15. The number of animals of a certain species was 100 000 at the end of 2010. The number N of the

animals t years after the end of 2010 can be represented by the following formula:

N = a(0.95)t, where a is a constant.

(a) Find the value of a.

(b) Find the number of the animals at the end of 2015.


16. The course fee $V of a training course t years after 2014 can be represented by the following formula:

V = 20 000kt, where k is a positive constant.

It is known that the course fee in 2016 is $22 050.


(b) Find the course fee of the training course in 2018.


17. Mary carries out a diet plan. Her weight W kg t months after the beginning of the diet plan can be

represented by the following formula:

W = 80pt, where p is a constant.

It is known that the weight of Mary after 1 month is 78.4 kg.

(a) Find the value of p.

(b) Mary claims that she can lose more than 20 kg within 12 months. Do you agree? Explain your

answer.

Level 2

18. The figure shows the graph of y = 6x. Sketch the graph of y = 6−x in

the same figure.

19. The figure shows the graphs of y = 4.5−x and y = 1.5−x.

(a) Which curve, C1 or C2, represents the graph of y = 4.5−x?

(b) Sketch the graph of y = 4.5x in the same figure.

23

y

x O

C3 C4 C5

P

y

x O

y = ax y = bx

y

x O

y = bx y = 4x

y = ax

20. The three curves in the figure represent the graphs of the exponential

functions y = 0.35x, y = 2.5x and y = 5−x.

(a) Complete the table below.

Curve Corresponding exponential function

C3

C4

C5

(b) Write down the coordinates of the point P in the figure.

(c) Sketch the graph of y =

x

5

2in the figure.

21. The figure shows the graphs of the exponential functions y = ax and

y = bx, where a and b are constants. The graph of y = bx is the image

of the graph of y = ax when reflected in the y-axis. If the difference

between a and b is greater than 4, suggest a pair of possible values of

a and b. Explain your answer.

22. The figure shows the graphs of the exponential functions y = ax, y = bx

and y = 4x, where a and b are constants. The graph of y = ax is the

image of the graph of y = 4x when reflected in the y-axis.


(b) Is the value of b greater than4

1? Explain your answer.

23. A researcher finds that the number N of a kind of fish in a pond under some controlled conditions after

t months can be represented by the following formula:

N = a(1.06)t, where a is a constant.

At the beginning of the research, the number of that kind of fish in the pond is 50 000.


(b) Find the number of that kind of fish in the pond 3 months from the beginning of the research.

(c) Find the number of that kind of fish increased from t = 2 to t = 5.

(Give the answers correct to the nearest integer if necessary.)

24

24. The value of a smart phone is $4 645 three months after it is on sale. The value $P of the smart phone

t months after it is on sale can be represented by the following formula:

P = 1 000 + k(0.9)t, where k is a constant.


(b) Find the depreciation of the smart phone after 1 year.


25. The value of a car in 2014 is $200 000. Its value $V after t years can be represented by the following

formula:

V = 200 000at, where a is a constant.

It is known that the value of the car after 1 year is $186 000.


(b) Find the decrease in the value of the car from 2014 to 2018.


26. In a factory, when x thousand of a kind of product are produced in a day, the amount Q units of a

particular pollutant in the air can be estimated by the formula:

Q = 100kx, where k is a positive constant.

It is known that the amount of the pollutant in the air is 121 units when 2 thousand of that kind of

product are produced in a certain day.


(b) Can the amount of the pollutant in the air be less than 100 units? Explain your answer.

27. After a new ride has been in operation for k months, the monthly number N of visitors in a theme park

can be represented by the following formula:

N = p + 80 000(1.15)k, where p is a constant.

It is known that the monthly number of visitors is 120 000 when k = 2.

(a) Find the value of p.

(b) Is it possible that the monthly number of visitors in the theme park is less than 93 000? Explain

your answer.

28. The temperature T °C inside a balloon after t hours can be represented by the following formula:

T = 30 + 70at, where a is a positive constant.

It is known that the temperature of the air inside the balloon after 2 hours is 74.8°C.

(a) Find the temperature of the air inside the balloon after 3 hours.

(b) Can the temperature of the air inside the balloon be higher than 100°C? Explain your answer.

25

29. The value $V of a flat t years after 2010 can be represented by the following formula:

V = 1 500 000kt, where k is a constant.

It is known that the value of the flat in 2013 was $1 889 568.


(b) If the value of the flat increases by more than 40% from 2012 to 2016, then the owner will sell

the flat. Will the owner sell the flat? Explain your answer.

30. A dose of 80 mg of medicine is given to a patient. The amount A mg of the medicine in the body of

the patient after t hours can be represented by the following formula:

A = kat, where a and k are constants.


(b) It is given that the amount of the medicine in the body of the patient after 1 hour is 56 mg.

(i) Find the value of a.

(ii) Is the amount of the medicine in the body of the patient less than 10 mg after 6 hours?

Explain your answer.

31. Ms Tang invested $500 000 in 2010. The amount $A obtained by her n (0 ≤ n ≤ 4) years after 2010 can

be represented by the following formula:

A = 500 000rn, where r is a positive constant.

It is known that the amount obtained by Ms Tang in 2012 was $605 000.

(a) Find the value of r.

(b) Find the amount obtained by Ms Tang in 2014.

(c) In 2015, Ms Tang will invest the amount $B obtained in (b) with a new offer. The amount $P

obtained by her t years after 2015 can be represented by the following formula:

P = B(1.2)t

Ms Tang claims that she will obtain an amount more than $1 800 000 in 2020. Do you agree?


32. The carbon dioxide level N (in suitable units) in cinema A t hours from the beginning of a movie can

be represented by the formula:

N = P(1 − 0.4t) + 200, where P is a constant.

It is known that the carbon dioxide level in cinema A is 440 units 1 hour from the beginning of the

movie.

(a) Find the value of P.

(b) The carbon dioxide level N1 (in suitable units) in cinema B t hours from the beginning of a movie

can be represented by N1 = 300 − 100(0.16)t. Is it possible that the carbon dioxide level in cinema

B is greater than that in cinema A? Explain your answer.

26

Answers

Consolidation Exercise 7B

1. (a) increases (b) decreases

(c) decreases (d) increases

(e) decreases (f) decreases

2. (a) y =

x

3

4 (b) y =

x

4

1

(c) y = 8x (d) y = 9x

(e) y =

x

3

10 (f) y = 0.4x

3. C1: y = 3x, C2: y = 7x

4. C3: y = 8x, C4: y =

x

2

1

5. C5: y = 0.3x, C6: y =

x

5

3

6. C7: y = 0.7x, C8: y = 3−x, C9: y = 3.5x

7. (b) 1

(c) (i) y ≥ 1 (ii) 0 < y < 1

8. (b) (0 , 1)

(c) (i) y ≥ 1 (ii) 0 < y < 1

9. (a) x −2 −1 0 1 2

y 4 2 1 0.5 0.25

(c) (0 , 1)

(d) (i) x > 0 (ii) x ≤ 0

10. $167 042

11. (a) $2 500 (b) $1 840

12. (a) 20 cm (b) 21.2 cm

13. 6 882

14. (a) 5 000 (b) 5 000

(c) 2 952

15. (a) 100 000 (b) 77 378

16. (a) 1.05 (b) $24 300

17. (a) 0.98 (b) no

19. (a) C2

20. (a) Curve

Corresponding

exponential function

C3 y = 0.35x

C4 y = 5−x

C5 y = 2.5x

(b) (0 , 1)

21. a =5

1, b = 5 (or other reasonable answers)

22. (a) 4

1 (b) no

23. (a) 50 000 (b) 59 551

(c) 10 731

24. (a) 5 000 (b) $3 590

25. (a) 0.93 (b) $50 390

26. (a) 1.1 (b) no

27. (a) 14 200 (b) no

28. (a) 65.84°C (b) no

29. (a) 1.08 (b) no

30. (a) 80

(b) (i) 0.7 (ii) yes

31. (a) 1.1 (b) $732 050

(c) yes

32. (a) 400 (b) no

27

F4B: Chapter 8A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)



(Full Solution)

28




___________ ( )




___________ ( )



___________ ( )



_________

29


Objective: To understand the definition of logarithms.

Meaning of Logarithms

Let a > 0 and a ≠ 1.

(a) If x = ay, then y = loga x.

(b) If y = loga x, then x = ay.

(c) log10 x (or log x) represents the common logarithm of x.


Find the value of log4 64. ∵ 43 = 64 ∴ log4 64 = 3

Find the value of log2 16.

∵ 2( ) = 16 ∴ log2 16 =

Find the values of the following logarithms. [Nos. 1–3] �Ex 8A: 3

1. log6 1 2. log9 3 3. log381

1

∵ 6( ) = 1

∴ log6 1 =


If log7 x = 2, find the value of x. ∵ log7 x = 2 ∴ x = 72

= 49

If log5 x = 3, find the value of x.

∵ log5 x = 3 ∴ x = ( )( )

=

In each of the following, find the value of x. [Nos. 4–6] �Ex 8A: 4

4. log4 x = 1 5. x

3

1log = 3 6. log9 x =2

1

∵ log4 x = ( )

∴ x = ( )( )

=


7. It is given that loga2

1=

2

1. Find the value of a.

ma

1= a–m

a = 2

1

a

�

Base = 4 Base = 2

Base = ( )

� loga x is called the logarithm of x with base a.

� loga x is meaningful only when a > 0,

a ≠ 1 and x > 0.

x = ay y = loga x

30

4B Lesson Worksheet 8.1B (Refer to Book 4B P.8.5)

Objective: To use the properties of logarithms to evaluate and simplify expressions.

Properties of Logarithms

Let a, b > 0, a, b ≠ 1 and x, y > 0.

(a) loga ax = x (b) loga 1 = 0 (c) loga a = 1

(d) loga xy = loga x + loga y (e) loga

y

x= loga x – loga y

(f) loga xk = k loga x (k is any real number) (g) loga x =

a

x

b

b

log

log(formula for change of base)


Find the value of log5 3 + log5

3

1.

log5 3 + log5

3

1= log5

×

3

13 � loga x + loga y = loga xy

= log5 1

= 0 � loga 1 = 0

Find the value of log2

1+ log 2.

log2

1+ log 2 = log [( ) × ( )]

= log ( )

=

Without using a calculator, find the values of the following. [Nos. 1–10]

1. log10 2 + log10 5 2. log3 81 – log3 9 �Ex 8A: 5, 6

= log10 [( ) × ( )] = log3

)(

)( � loga x – loga y = loga

y

x

= log10 ( ) = log3 ( )

= � loga a = 1 = log3 ( )( )

= � loga ax = x

3. log4 8 + log4 2 4. log10 4 – log10 40

= =

5. 9

2

42

7log

7log 6.

2log

32log �Ex 8A: 7

= 7log)(

7log)(

2

2 � loga xk = k loga x =)(log

)(log

= =

32 = 2 ( )

�

log4 4x = x

31

7. log 3 × log3 10 8. log2 3 × log3 2

= log 3 ×)(

)( � loga x =

a

x

b

b

log

log =

=

9. log4 5 × log5 16 10. log5 9 × log3 25

= =

Simplify the following expressions, where x > 0 and x ≠ 1. [Nos. 11–14]

11. 3

4

log

log

x

x 12. log4 x

3 – 3 log4 x �Ex 8A: 8

= =

13. log24

6x

– 6 log2 x 14. (log x)(logx 100)

= =


15. (a) Express 75 as a product of 3 and 5.

(b) Let log 3 = a and log 5 = b. Express log 75 in terms of a and b.

32


8 Logarithmic Functions


Level 1

1. Convert each of the following expressions into logarithmic form.

(a) 16 = 24 (b) 100 = 102 (c) 1 = 80 (d) 27

1= 3–3

(e) 5 = 3

1

125 (f) 24 = x4 (g) m = 3n (h) 12 = ab

2. Convert each of the following expressions into exponential form.

(a) 3 = log2 8 (b) 3 = log 1 000 (c) 0 = log7 1 (d) –4 = log2 16

1

(e) 6

1= log64 2 (f) 5 = log3 x (g) c = logd 5 (h) 8 = loga b

3. Without using a calculator, find the values of the following logarithms.

(a) log2 25 (b) log3 9 (c) log5 1 (d) log 100

(e) log27 3 (f) log3

9

1 (g) logx x3 (h) y

ylog

4. In each of the following, find the value of x.

(a) log x = 5 (b) log5 x = 4 (c) log1.2 x = 2 (d) log4 x =2

1−

(e) logx 64 = 3 (f) logx

25

1= 2 (g) logx 9 = –2 (h) logx 3 =

4

1−

5. Without using a calculator, find the values of the following.

(a) log6 4 + log6 9 (b) log4 2 + log4 32 (c) log 25 + log 4 (d) log 20 + log 0.5

(e) log5 0.4 + log5 12.5 (f) log3 6 + log3

2

9 (g) log7

2

7 + log7 14 (h) log4

16

3+ log4

12

1


(a) log2 40 – log2 10 (b) log5 100 – log5 4 (c) log 300 – log 30 (d) log6 7 – log6 42

(e) log8 3 – log8 192 (f) log4 56 – log4

2

7 (g) log

5

1– log 20 (h) log3 0.04 – log3 1.08


(a) 3log

27log

2

2 (b) 25log

5log

4

4 (c) 4log

8log (d)

81log

27log

(e) 64log

16log

3

3 (f) 9log

1log

5

5 (g) 3.0log

09.0log

8

8 (h)

25log

2.0log

6

6

33

8. Simplify the following expressions, where x > 0 and x ≠ 1.

(a) 4 log2 x – log2 x4 (b) log3 x + log3

x

3 (c) log (4x) – log (40x) (d)

25

55

log

log

x

x

(e)

24

4

1log

log

x

x (f)

x

x

log

1log

(g) (log5 x)(logx 25) (h) (logx 8)(log4 x)

9. If loga x = 3, find the values of the following expressions.

(a) loga x4 (b) loga x

–6 (c) logx a (d) xa2log

10. If log 5 = m, express the following in terms of m.

(a) log 125 (b) log 0.2 (c) log

5

1 (d) log 500

(e) log 2 (f) log 2.5 (g) log 40 (h) log25 100

11. If log 3 = a and log 5 = b, express the following in terms of a and b.

(a) log 15 (b) log 0.6 (c) log5 3 (d) 5log3

1

12. Use a calculator to find the values of the following, correct to 3 significant figures.

(a) log 18 (b) log 0.37 (c) log

7

32 (d) log

5 36

13. Use a calculator to find the values of the following, correct to 2 decimal places.

(a) log3 8 (b) log6 2.5 (c) log5 18 (d) 7

22log

6

14. Determine whether each of the following is true (T) or false (F).

(a) log3 43 = 4 (b) log 2 + log3 5 = 1

(c) log5 0 = 1 (d) log4 x = –2 has no solutions.

(e) log (–10) is undefined. (f) If x ≠ 0, then log x2 = 2 log x.

(g) log(–2) (–8) = 3 (h) k

x

1log = –k log x for x > 0.

Level 2


(a) log4 3 54 (b) log

5 01.0

1

(c) log3 3 9

27 (d) log5

4 3

2

5

25

−

−

34


(a) log4 4 8 + log4

4 2 (b) log 2–3 + log

25

2

(c) log3 (4 ⋅ 3

5

9 ) – log3 12 (d) log6 3 108 – log6

3 3

(e) 2

1log5 45 + log5 3

27

25 (f)

2

3log 90 – log 0.27


(a) log 12 + log 20 – log 24 (b) log4 3 – log4 75 + log4 100

(c) log2 0.4 + log2

3

8+ log2 60 (d) 4 log5 10 – 2 log5 12 + log5 45

(e) 3 log3 15 + log3 0.6 – 2 log3 5 (f) log6 120 – log6 150 – 2 log6

3

1


(a) 6log

12log18log + (b)

96log6log

16log

33

3

−

(c) 24log4log

54log4log

−

+ (d)

50log20log

2.0log240log

66

66

+

−

(e) (log3 37 )(log7 9) (f)

34

9

1log (log3

52 )

(g) 27log

9log

4

2 (h) log5 3 –25log

45log

3

3

19. Simplify the following expressions, where x > 1.

(a) 3

5

log

log4log

x

xx + (b)

xx

x

log5log

log2

3

−

(c) 34

2

log2

1log2

log3log4

1

xx

xx

−

+ (d)

1log

136log

6

6

+

−

x

x

(e) x

x

3

23

log1

3log1

+

+ (f)

x

x

4

22

log

log

(g) 2 log4 x + 3 log8

x

1 (h)

25

25

25

)5(log

5log)(log

x

xx +

20. Simplify the following expressions, where x, y > 1.

(a) yx

xxy

loglog

log2log 3

+

+ (b)

3

2

log

log2log

y

xyyx −

(c) y

xy

x

x

log21

log 2

+ (d) ))(log(log 52

43 yxxy

(e) 3loglog 3 yy xx− (f) (log x)(logx y

2)(logy 1 000)

(g) )100)(log(log

log

x

x

y

y (h)

))(log(log

log)(log2

322

yxxy

yxy

xx

xx +

35

21. If log 3 = a and log 5 = b, express the following in terms of a and b.

(a) log 75 (b) log 225 (c) log 0.12 (d) 125log

27log

(e)

5

9log

5log (f) log25 15 (g) 50log

3

1 (h) log

9

2

22. If log 2 = c and log 7 = d, express the following in terms of c and d.

(a) log 56 (b) log

8

49 (c) log 1.75 (d)

8log

49log

(e) 14log2 (f) log28 98 (g) log

5

7 (h)

2

35log14

23. If log x = m and log y = n, express the following in terms of m and n, where x, y > 0 and x, y ≠ 1.

(a) log

xy

100 (b) log

3

2

x

y (c) logx xy + logy

y

x (d) 4 23

2log yxx

24. If 10p = 2 and 10q = 7, express log 280 in terms of p and q.

25. If 10x = 2 and 5y = 3, express log 6 in terms of x and y.

26. In each of the following, express y in terms of x.

(a) 2 log x = 3 log y (b) log4 (x2y) = 3

(c) logy x = 5 (d) logx y = 8(logy x)2

(e) log (x + y) = log x + log y (f) log (x – y) = log y – 2 log x

(g) 4 log x + log y = 2 (h) 3 log5 x – log5 y = 2

27. Use a calculator to find the value of x in each of the following, correct to 3 significant figures.

(a) log x = 3.14 (b) log x =70

22−

(c) log4 x = 1.6 (d) 2log3

−=x

28. (a) Show that xaxa =log , where a, x > 0 and a ≠ 1.

(b) Find the value of 13log13log 66 32 ⋅ without using a calculator.

(c) Simplify k

m

k log

)(log log

, where k > 0, k ≠ 1 and m > 1.

29. (a) Show that alog x = xlog a, where a, x > 0.

(b) Using the result of (a), find the values of the following.

(i) (5log 8)(8log 2)

(ii) (3log 25)(2log 9)

36

30. Find the value of (log2 4)(log4 6)(log6 8)…(log14 16) without using a calculator.

31. If x = log 5 + (log 2)i and y = log 4 – (log 25)i, where i = 1− , find the value of 2x – iy.

32. If z = log (x + 2) + [log (x2 + 5x + 5)]i is a real number, where i = 1− , find the values of x and z.

33. α and β are the two roots of the equation x2 – 100x + 10 = 0. Find the values of the following

expressions.

(a) log α2 + 2 log β

(b) logα + β

β

1+ logα + β

α

1

(c) log20 α3 + 3 log20 2β

(d) log (100 – α) + log (100 – β)

34. The minimum value of the quadratic function f(x) = (log4 a)x2 − 4x + 3 is –5. Find the value of a.

35. (a) By taking logarithms on both sides of y = abx, show that

log3 y = log3 a + x log3 b.

(b) The graph in the figure shows the linear relation between log3 y

and x. If y = abx, find the values of a and b.

36. The graph in the figure shows the linear relation between log y and log

x. If y = cxn, find the values of c and n.

(Leave the radical sign ‘√’ in the answers.)

37. The graph in the figure shows the linear relation between log8 y and x.

Express y in terms of x.

38. The figure shows the graph of y = mnx, where m and n are constants.

Write down the relation between x and log2 y.

4

2 0

log3 y

x

–3 0

log y

log x 30°

y

4

0x

y = mnx

(–2 , 16)

–2

3 0

log8 y

x

37

Answers


1. (a) 4 = log2 16 (b) 2 = log 100

(c) 0 = log8 1 (d) –3 = log327

1

(e) 3

1= log125 5 (f) 4 = logx 24

(g) n = log3 m (h) b = loga 12

2. (a) 8 = 23 (b) 1 000 = 103

(c) 1 = 70 (d) 16

1= 2–4

(e) 2 = 6

1

64 (f) x = 35

(g) 5 = d c (h) b = a8

3. (a) 5 (b) 2 (c) 0 (d) 2

(e) 3

1 (f) –2 (g) 3 (h) 2

4. (a) 100 000 (b) 625

(c) 1.44 (d) 2

1

(e) 4 (f) 5

1

(g) 3

1 (h)

81

1

5. (a) 2 (b) 3 (c) 2 (d) 1

(e) 1 (f) 3 (g) 2 (h) –3

6. (a) 2 (b) 2 (c) 1 (d) –1

(e) –2 (f) 2 (g) –2 (h) –3

7. (a) 3 (b) 2

1 (c)

2

3 (d)

4

3

(e) 3

2 (f) 0 (g) 2 (h)

2

1−

8. (a) 0 (b) 1 (c) –1 (d) 2

5

(e) 2

1− (f) –2 (g) 2 (h)

2

3

9. (a) 12 (b) –18 (c) 3

1 (d)

2

3

10. (a) 3m (b) –m

(c) 2

m− (d) m + 2

(e) 1 – m (f) 2m – 1

(g) 3 – 2m (h) m

1

11. (a) a + b (b) a – b (c) b

a (d)

a

b−

12. (a) 1.26 (b) –0.432 (c) 0.660 (d)

0.311

13. (a) 1.89 (b) 0.51 (c) 0.90 (d) 1.28

14. (a) F (b) F

(c) F (d) F

(e) T (f) F

(g) F (h) T

15. (a) 3

5 (b)

5

2 (c)

6

5 (d)

4

13−

16. (a) 2

1 (b) –2 (c)

3

7

(d) 3

2 (e)

6

7 (f)

2

7

17. (a) 1 (b) 1 (c) 6

(d) 5 (e) 4 (f) 2

5

18. (a) 3 (b) –1 (c) –3 (d) 1

(e) 3 (f) 6

5− (g)

3

8 (h)

2

1−

19. (a) 3 (b) 9

1− (c) –2 (d) 1

(e) 2 (f) 4 (g) 0 (h) 1

20. (a) 3 (b) –3 (c) 1 (d) 6

5

(e) 0 (f) 6 (g) 4

1 (h) 1

21. (a) a + 2b (b) 2a + 2b

(c) a – 2b (d) b

a

(e) ba

b

24 − (f)

b

ba

2

+

(g) a

b 1+− (h)

2

1(1 – 2a – b)

22. (a) 3c + d (b) 2d – 3c

(c) d – 2c (d) c

d

3

2

(e) c

dc

2

+ (f)

dc

dc

+

+

2

2

(g) c +2

d– 1 (h)

)(2

21

dc

dc

+

+−

23. (a) 2 – m – n (b) 2n –3

m

(c) m

n

n

m+ (d)

m

nm

8

23 +

24. 2p + q + 1 25. x + y – xy

38

26. (a) y = 3

2

x (b) y =2

64

x

(c) y = 5

1

x (d) y = x2

(e) y =1−x

x (f) y =

12

3

+x

x

(g) y =4

100

x (h) y =

25

3x

27. (a) 1 380 (b) 0.485 (c) 9.19 (d)

0.460

28. (b) 13 (c) log m

29. (b) (i) 8 (ii) 9

30. 4 31. 0

32. x = –1, z = 0

33. (a) 2 (b) 2

1− (c) 3 (d) 1

34. 2 35. (b) a = 81, b

=9

1

36. c = 310 , n =3

1 37. y = 22x – 6

38. log2 y = 2 – x

39

F4B: Chapter 8B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 8


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )



___________ ( )



_________

40


Objective: To understand the properties of logarithmic functions and their graphs.

Logarithmic Functions

Let x and y be variables, a be a constant, where a > 0 and a ≠ 1. For x > 0, y = loga x is called a logarithmic

function with base a.

Properties of the Graphs of Logarithmic Functions

Range of a a > 1 0 < a < 1

Graph of

y = loga x

Common

properties

(a) The graph does not cut the y-axis. It lies on the right of the y-axis.

(b) The graph cuts the x-axis at the point (1 , 0).

(c) The graphs of y = x

a

1log and y = loga x are the images of each other when

reflected in the x-axis.

(d) The graphs of y = ax and y = loga x are the images of each other when reflected in

the line y = x.

Different

properties

When the value of x increases, the value

of y increases and the rate of increase of y

decreases.

When the value of x increases, the value

of y decreases and so does the rate of

decrease of y.

1. 2.




3. The two curves in the figure represent the graphs of the

logarithmic functions y = log5 x and y = x

5

1log . Write down

the corresponding logarithmic function for each of C1 and C2.

C1:

C2:

`

y = log3 x

x O P

y

x O P

y

y = x

6

1log

�

C1

y

x O

C2

� Ex 8B: 4, 5

0 <6

1< 1 3 ( < / > ) 1

41


logarithmic functions y = log x and y = log6 x. Write down

the corresponding logarithmic function for each of C1 and C2.

C1:

C2:


functions y = 8x and y = log8 x. Write down the corresponding


C1:

C2:

6. The figure shows the graph of y = log4 x. 7. The figure shows the graph of xy

3

1log= .

Sketch the graph of y = log4

1 x in the same figure. Sketch the graph of y = log3 x in the same figure.

8. The figure shows the graph of y = 6x. 9. The figure shows the graph of y = log4 x.

Sketch the graph of y = log6 x in the same figure. Sketch the graph of y = 4x in the same figure.


10. The two curves in the figure represent the graphs of the logarithmic

functions y = log0.2 x and y = log0.5 x.

(a) Write down the corresponding function for each of C1 and C2.

(b) The straight line x = k passes through the point of intersection of

C1 and C2. Find the value of k.

x O

y y = x

y = log4 x

x O

y y = 6x y = x

x

C1

O

y

C2

� Ex 8B: 6

� Ex 8B: 7, 8

C2 y

O

C1

x

y = log4 x

y

x O

y

x O

y = x

3

1log

x

y

O

C1

C2

42




Level 1

1. Each of the following graphs of functions is reflected in the x-axis. What is the function represented

by each graph obtained?

(a) y = log x (b) y = log0.5 x

(c) y = x

6

1log (d) y = x

2

7log

Each of the following graphs of functions is reflected in the line y = x. What is the function represented by

each graph obtained? [Nos. 2–3]

2. (a) y = 5x (b) y = 1.5x

(c) y =

x

3

5 (d) y =

x)24(

3. (a) y = log4 x (b) y = log0.7 x

(c) y = x

4

3log (d) y = logπ x


functions y = log4 x and y = log0.2 x. Write down the corresponding

logarithmic functions for C1 and C2.

5. In the figure, the two curves represent the graphs of the logarithmic

functions y = log3 x and y = log6 x.

(a) Write down the corresponding logarithmic functions for C3 and

C4.

(b) C3 and C4 intersect at a point P. Find the coordinates of P.


functions y = log0.1 x and y = log0.4 x.

(a) Write down the corresponding logarithmic functions for C5 and

C6.

(b) The straight line x = k passes through the point of intersection of

C5 and C6. Find the value of k.

x

y

O C5

C6

x

y

O

C1

C2

x

y

O

C3

C4

P

43

According to the graph given in each of the following, sketch the graph of the required function in the

same given graph. [Nos. 7–9]

7. (a) y = x

4

1log (b) y = log6 x

8. (a) y = log3 x (b) y = log0.3 x

9. (a) y = 2x (b) y = 0.7x

3

2

1

x

–1

–2

–2 –1

y

0 1 2 3

y = x

y = 0.3x

3

2

1

x

–1

–2

–2 –1

y

0 1 2 3

y = x

y = 3x

3

2

1

x

–1

–2

–2 –1

y

0 1 2 3

y = x

y = log0.7 x

3

2

1

x

–1

–2

–2 –1

y

0 1 2 3

y = x

y = log2 x

2

1

x

–1

–2

0 1 2 3 4

y

y = x

6

1log

2

1

x

–1

–2

0 1 2 3 4

y

y = log4 x

44

Level 2

10. In each of the following, the graph of the function is reflected in the x-axis, and the image is then

reflected in the line y = x. What is the function represented by each of the final graphs?

(a) y = log2.5 x

(b) y = log0.8 x

11. In each of the following, the graph of the function is reflected in the line y = x, and the image is then

reflected in the y-axis. What is the function represented by each of the final graphs?

(a) y = x

7

2log

(b) y = log1.6 x

12. The four curves in the figure represent the graphs of the logarithmic functions y = log0.4 x, y = log0.6 x,

y = log3 x and y = log5 x.

(a) Write down the corresponding logarithmic functions for C1, C2, C3 and C4.

(b) The straight line y = 2x + k passes through the point of intersection of the four curves. Find the

value of k.

13. The three curves in the figure represent the graphs of the functions y = log0.2 x, y = 0.5x and y

= 12

1 2 +x .

(a) Write down the corresponding functions for C5, C6 and C7.

(b) C5 cuts the y-axis at the point P(0 , k). Angela claims that C7 cuts the x-axis at the point Q(k , 0).

Do you agree? Explain your answer.

O

C5

x

y

C6

C7

x

y

C1

C2

C3

C4

O

45

14. The graph of y = logp x is the image of the graph of y = logq x when reflected in the x-axis, where p

and q are constants with p > q. If the sum of p and q is less than 3, suggest a pair of possible values of

p and q. Explain your answer.

15. The graph of y = log(k – 1) x is the image of the graph of y = (k – 3)2x when reflected in the line y = x,

where k is a constant. Find the value of k.

16. The figure shows the graphs of y = logm x, y = logn x and y = log3 x, where m and n are constants. The

graph of y = logm x is the image of the graph of y = log3 x when reflected in the x-axis.

(a) Find the value of m.

(b) Write down two possible values of n. Explain your answer.

17. (a) According to the graph of y = 5x, sketch the graph of y = log5 x in the same given graph.

(b) From the graph of y = log5 x obtained,

(i) write down the range of values of y when x ≥ 1,

(ii) write down the range of values of x when y < 0.

(c) Using the result of (a), sketch the graph of y = log0.2 x in the same given graph.

3

2

1

x

–1

–2

–2 –1

y

0 1 2 3

y = x

y = 5x

x

y

O

y = log3 x

y = logm x

y = logn x

46

18. In the figure, C is the image of the graph of y = 3x when reflected in the line L: y = x.

(a) Write down the corresponding function for C.

(b) The y-coordinate of the point G is 2. The point H is the image of G when reflected in the line y =

x.

(i) Find the equation of the straight line passing through G and H.

(ii) Is L the perpendicular bisector of GH? Explain your answer.

19. The two curves C1 and C2 in the figure represent the graphs of the logarithmic functions y = log4 x and

y = log2 x.

(a) Write down the corresponding functions for C1 and C2.

(b) It is given that P(8 , k) is a point on C1.

(i) Find the value of k.

(ii) Q and R are two points on C2. Q has the same y-coordinate as P and PQ ⊥ PR. Find the

coordinates of Q and R.

y

x

C1

C2

O

L: y = x

x

y

O

C

y = 3x

G

H

47

Answers

Consolidation Exercise 8B 1. (a) y =

10

1log x (b) y = log2 x

(c) y = log6 x (d) y =7

2log x

2. (a) y = log5 x (b) y = log1.5 x

(c) y =3

5log x (d) y =24

log x

3. (a) y = 4x (b) y = 0.7x

(c) y =

x

4

3 (d) y = πx

4. C1: y = log4 x, C2: y = log0.2 x

5. (a) C3: y = log3 x, C4: y = log6 x

(b) (1 , 0)

6. (a) C5: y = log0.1 x, C6: y = log0.4 x

(b) 1

10. (a) y = 0.4x (b) y = 1.25x

11. (a) y =

x

2

7 (b) y = 0.625x

12. (a) C1: y = log3 x, C2: y = log5 x,

C3: y = log0.4 x, C4: y = log0.6 x

(b) –2

13. (a) C5: y = 12

1 2 +x , C6: y = 0.5x,

C7: y = log0.2 x

(b) yes

14. p = 2, q =2

1 (or other reasonable answers)

15. 5

16. (a) 3

1

(b) 4

1,

5

1 (or other reasonable answers)

17. (b) (i) y ≥ 0 (ii) 0 < x < 1

18. (a) y = log3 x

(b) (i) x + y – 11 = 0

(ii) yes

19. (a) C1: y = log2 x, C2: y = log4 x

(b) (i) 3

(ii) Q(64 , 3), R

2

3 , 8

48

F4B: Chapter 8C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

49



(Full Solution)

Maths Corner Exercise 8C Level 1



___________ ( )




___________ ( )

Maths Corner Exercise 8C Multiple Choice


___________ ( )



_________

50

4B Lesson Worksheet 8.3A & B (Refer to Book 4B P.8.26)

Objective: To solve exponential equations and logarithmic equations.

Review: Properties of Logarithms

Let a > 0, a ≠ 1 and x, y > 0. In each of the following, fill in the blanks. [Nos. 1–4]

1. If x = ay, then y = log( ) ( ). 2. If y = loga x, then x = ( )( ).

3. loga x4 = ( ) log( ) ( ) 4. yalog = ( ) log( ) ( )

Exponential Equations

An equation involving unknown index (indices) is called an exponential equation.


(a) If ax = ay, then x = y. (b) If ax = y, then x = loga y =a

y

log

log.


Solve 43x + 1 = 45 – x.

43x + 1 = 45 – x ∴ 3x + 1 = 5 – x

4x = 4

x = 1

Solve 54x – 2 = 54 + 2x.

54x – 2 = 54 + 2x ∴ =

=

Solve the following exponential equations. [Nos. 5–6]

5. 2x – 2 = 2 6. 32 – 3x =9

1 �Ex 8C: 1–6

2x – 2 = 2( ) =

∴ =

=


Solve 3x = 11 and give the answer correct

to 3 significant figures.

3x = 11

log 3x = log 11 �

x log 3 = log 11 � loga xk = k loga x

x =3log

11log

= 2.18, cor. to 3 sig. fig.

Solve 8x = 16 and give the answer correct


8x = 16

log 8x = ( )

( ) log 8 = ( )

x =)(

)(

=

�

Rewrite9

1as a

power of 3.

� When a > 0 and a ≠ 1, we can obtain x = y from ax = ay.

y = y( )

� Change both sides of the equation to powers of the same base.

Take common logarithms on both sides of the equation.

Take common logarithms on both sides of the equation.

51

Solve the following exponential equations and give the answers correct to 3 significant figures. [Nos. 7–8]

7. 19x = 1.9 8. 52x = 15 �Ex 8C: 7–12

Logarithmic Equations

An equation involving unknowns in logarithms is called a logarithmic equation.


(a) If loga x = y, then x = ay. (b) If loga x = loga y, then x = y.


Solve log5 x = 3 and log2 (y + 4) = log2 3. Solve log2 x = 4 and log3 (y + 7) = log3 (3 – y).

log5 x = 3 ∴ x = 53

= 125

log2 (y + 4) = log2 3 ∴ y + 4 = 3

y = –1

log2 x = 4 ∴ ( ) = ( )( )

=

log3 (y + 7) = log3 (3 – y) ∴ ( ) = ( )

=

Solve the following logarithmic equations. [Nos. 9–12]

9. log3 x = –2 10. log4 x = –3 �Ex 8C: 13–18

11. log (x + 3) = 4 log 3 12. log4 (2x + 1) = 2 log4 5


13. If log5 8x = 2 log25 3y, find x : y.

Change both sides of the equation to logarithms of the same base.

52



� Consolidation Exercise 8C

Level 1

Without using logarithms, solve the following exponential equations. [Nos. 1–6]

1. 42x = 43 + x 2. 62x + 3 = 64 – x

3. 34x – 1 = 27 4. 52x + 1 =3 25

5. 103 – x = 3 100 6. 23 + 2x =5 4

1

Solve the following exponential equations and give the answers correct to 3 significant figures. [Nos. 7–

12]

7. 5x = 8 8. 8x = 1.6

9. 1.2x = 6 10. 3x = 7

11. 6x + 2 = 18 12. 41 – 2x =7

1


13. log3 (x – 2) = 2 14. log7 (4 – x) = 0

15. 3 log4 x – 2 = 1 16. 3 log8 x + 1 = 0

17. log5 (x + 2) = 3 log5 2 18. log (8x) – 2 log 4 = 0

19. log8 (1 + 2x) = –2 log8 3 20. log6 (2 – 3x) + 4 log6 2 = 0

21. The value $P of an antique vase after n years is given by the formula:

P = 3 000 × 1.06n

If the value of the antique vase will be $6 030 after k years, find the value of k.


22. One end of a metal rod is heated to 400°C. The temperature T°C at a point d m from that end point on

the metal rod is given by the formula:

log T = A – 0.8d, where A is a constant.

(a) Find the value of A.

(b) Find the temperature at the point 0.3 m from that end point on the metal rod.

(Give the answers correct to 3 significant figures.)

53

Level 2

Without using logarithms, solve the following exponential equations. [Nos. 23–34]

23. 92x + 1 = 31 – x 24. 83 + x =124

1−x

25.

x−

2

5

1= 25x + 2 26. 36 =

x

x

−

+

3

2

36

6

27. 24 – x ⋅ 4x + 3 = 32 28. 27x + 1 ⋅ 92 – 2x = 3

29. 6x + 1 – 6x =36

5 30. 2x + 2 + 2x = 210

31. 5x – 52 + x + 120 = 0 32. 33 + 2x – 9x + 1 – 54 = 0 33. 3x ⋅ 6x + 2 = 9x + 1 34. 24x + 3 + 42x + 28 = 16x + 1

Solve the following exponential equations and give the answers correct to 3 significant figures.

[Nos. 35–42]

35. 5(3x + 1) = 12 36. 2(10x + 1) – 0.5 = 0

37. 122

7−x

= 11 38. 4x = 7x – 2

39. 6x ⋅ 4x + 2 = 24 40. 53x = 8(33 – x)

41. 8x + 2 – 3(23x + 1) = 9 42. 2x + 2 ⋅ 3x + 1 = 4x


43. log100 x + log x2 = 5 44. log4 x – 2 log8 x = –1

45. x

x

3

9

log

27log= 2 46. log7 (5 – x) – log7 (3 + x) = 1

47. log (2x + 3) + 1 = log (4x – 2) 48. log3 (2x + 1) = log3 (7x – 1) – 1 49. 2 log25 (x – 3) + 1 = log5 (3x + 1) 50. log2 (x + 3) + log4 (x + 3) = 3 51. log (log x) = 0 52. log3 [log2 (log4 x)] = 1

53. Solve log4 (22x + 1 + 8) = x + 1.

54. (a) Solve log u = log (2u + 4) – 1.

(b) Using the result of (a), solve log 2x = log (2x + 1 + 4) – 1.

54

55. (a) It is given that 9 ≤ log k < 10.

(i) Find the range of values of k.

(ii) Hence, determine the number of digits of k.

(b) It is given that log 2 = 0.301, correct to 3 decimal places. Find the number of digits of 230 without

using a calculator.

56. The depreciation rate of a mobile phone is 20% every year. Find the minimum number of years

required for the mobile phone to be less than half of its original value.

57. Billy deposits $100 000 into a bank at an interest rate of 3% p.a. compounded monthly. At least how

many months later can Billy receive an amount more than $115 000?

58. Alfred deposits $30 000 into a bank at an interest rate of 12% p.a. compounded yearly. At the same

time, Cathy deposits $50 000 into a bank at an interest rate of 5% p.a. compounded yearly. At least

how many years later will the total amount received by Alfred be more than that received by Cathy?

59. A reservoir holds 3 000 000 m3 of water initially. The volume of water in the reservoir decreases by

k% every month, where k is a constant. The volume of water in the reservoir after 6 months is 2 205 000

m3.

(a) Find the value of k, correct to the nearest integer.

(b) The reservoir is said to be emergent if the volume of water in the reservoir is less than 1 000 000 m3.

At least how many months later will the reservoir be emergent?

60. The total cost C (in thousand dollars) of producing n electronic devices in a factory can be estimated

by the following formula: C = a + log3 (n + 1), where a is a constant.

The total cost of producing 80 electronic devices in the factory is $9 000.


(b) Find the number of electronic devices produced in the factory with a total cost of $12 000.

(c) 100 electronic devices are produced in the factory on a day and packed in box A. 50 electronic

devices are produced on another day and packed in box B. The production manager claims that

the cost of each electronic device in box A is less than half of that in box B. Do you agree?


61. An astronomer defined Scale A and Scale B to represent the

magnitudes of brightness of stars as shown in the table. S and T are

the magnitudes of brightness on Scale A and Scale B respectively,

while L is the relative brightness of the star. The magnitude of

brightness of star X is 2 on Scale A.

(a) Find the magnitude of brightness of star X on Scale B, correct to 3 significant figures.

(b) The relative brightness of another star Y is 10 times that of star X. Is the magnitude of brightness

of star Y half of that of star X on Scale B? Explain your answer.

Scale Formula

A S = 3 – log10 L

B T = 5 – log5 L

55

Answers

Consolidation Exercise 8C

1. 3 2. 3

1 3. 1 4.

6

1−

5. 3

7 6.

10

17− 7. 1.29 8. 0.226

9. 9.83 10. 0.886 11. –0.387 12. 1.20

13. 11 14. 3 15. 4 16. 2

1

17. 6 18. 2 19. 9

4− 20.

48

31

21. 12

22. (a) 2.60 (b) 230°C

23. 5

1− 24. –1 25. –6 26. 2

27. –5 28. 6 29. –2 30. 2

3

31. 1 32. 2

1 33. –2 34.

2

1

35. –0.203 36. –1.60 37. 0.174 38. 6.95

39. 0.128 40. 0.907 41. –0.896 42. –6.13

43. 100 44. 64 45. 3 46. –2

47. no real solutions 48. 4 49. 8

50. 1 51. 10 52. 65 536 53. 1

54. (a) 2

1 (b) –1

55. (a) (i) 109 ≤ k < 1010

(ii) 10

(b) 10

56. 4

57. 56 months

58. 8 years

59. (a) 5 (b) 22 months

60. (a) 5 (b) 2 186

(c) no

61. (a) 3.57 (b) no

56

F4B: Chapter 8D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)

Book Example 20


(Video Teaching)



(Full Solution)

Maths Corner Exercise 8D Level 1



___________ ( )

Maths Corner Exercise 8D Level 2



___________ ( )

Maths Corner Exercise 8D Multiple Choice


___________ ( )



_________

57


Objective: To solve problems related to the Richter scale.

Richter Scale

Magnitude M on the Richter scale = log A + K,

where A units represents the amplitude of the seismic wave measured and K is a constant.

1. An earthquake occurred in city X and the 2. An earthquake of magnitude 8 occurred in city B.

amplitude recorded was 40 units. Using the Using the formula M = log A + 4.24, find the

formula M = log A + 4.24, find the magnitude amplitude recorded, correct to 1 decimal place.

of the earthquake, correct to 1 decimal place. ( ) = log A + ( )

M = ( ) + 4.24 log A =

= ( ), cor. to 1 d.p.

∴ The magnitude of the earthquake was

( ).

3. An earthquake occurred in both city P and 4. Two earthquakes occurred in city X. The

city Q. The amplitudes recorded were 100 units amplitudes recorded in the first and the second

and 400 units respectively. Find the difference earthquakes were 810 units and 540 units

in the magnitudes of the earthquakes in these respectively. Find the difference in the

two cities. magnitudes of the two earthquakes. �Ex 8D: 3, 4

(Give the answer correct to 1 decimal place.) (Give the answer correct to 1 decimal place.)

Magnitude of the earthquake in city P

=

Magnitude of the earthquake in city Q

=

Difference in the magnitudes of the earthquakes

in these two cities

= ( ) – ( )

=

∴ The magnitudes of the earthquakes in these

two cities differ by ( ).

�

If loga x = y, then x = ay.

�Ex 8D: 1, 2

log M – log N =N

Mlog

58

5. The magnitude of an earthquake that occurred 6. The magnitudes of two earthquakes that occurred

in a city was 7 on the Richter scale. Later, an in city X and city Y were 7.2 and 4.8 respectively.

aftershock of magnitude 5 was recorded in the How many times was the amplitude of the

same city. How many times was the amplitude earthquake in city X as large as that in city Y?

of the first earthquake as large as that of the (Give the answer correct to the nearest integer.)

aftershock? �Ex 8D: 12, 13

Let the amplitude of the aftershock be A units. Let the amplitude of

( ) = log A + K be A units.

Let the required times be x.

The magnitude of the first earthquake can be

expressed as:

( ) = log xA + K

( ) = log ( ) + log ( ) + K

( ) = log ( ) + ( )

=

∴ The amplitude of the first earthquake was

( ) times as large as that of the aftershock.


7. An earthquake occurred in both city P and city Q. The amplitude recorded in city P was 360 units

more than that recorded in city Q. If the magnitudes of the earthquakes in the two cities differ by

1.5, find the amplitude recorded in city Q. (Give the answer correct to the nearest integer.)

log MN = log M + log N

59


Objective: To solve problems related to the intensity level of sound.

Intensity Level of Sound

For a sound of intensity I (W/m2), its intensity level D (dB) is:

D = 10 log0I

I, where I0 = 10–12 W/m2.

1. If the intensity of a sound is 10–7 W/m2, then 2. If the intensity of a sound is 10–2 W/m2, then

the intensity level of the sound the intensity level of the sound

= 10 log)(

)(dB =

= 10 log 10( ) – ( ) dB

= 10 log 10( ) dB

= 10 ⋅ ( ) dB

= dB

3. The intensity of the sound measured in a contruction site is 10–9 W/m2. Find the intensity level of the sound.


The intensity level of a sound produced by a bus

is 74 dB. Find the intensity of the sound, correct


Let the intensity of the sound be I W/m2.

74 = 10 log1210−

I

7.4 = log1210−

I ∴

1210−

I= 107.4

I = 10–12 × 107.4

= 10–4.6

= 2.51 × 10–5, cor. to 3 sig. fig. ∴ The intensity of the sound is 2.51 × 10–5 W/m2.

The intensity level of a sound produced by a

machine is 105 dB. Find the intensity of the sound,

correct to 3 significant figures.

Let the intensity of the sound be .

( ) = 10 log)(

)(

( ) = log)(

)( ∴

)(

)(= ( )( )

I =

�

�Ex 8D: 6

n

m

a

a= am – n

am ⋅ an = am + n

60

4. The intensity level of a sound produced by a 5. The intensities of the sound measured in a bank

vacuum cleaner is 57 dB. Find the intensity of and a karaoke lounge are 10–5 W/m2 and

the sound, correct to 3 significant figures. 10–3 W/m2 respectively. Find the difference in

�Ex 8D: 7 the intensity levels of the two places. �Ex 8D: 8

Difference in the intensity levels of the two places

=

−

)()(

log10)()(

log10 dB

= ( )( ) dB

6. The intensity of a sound produced by an instrument is 10–5.7 W/m2. If the intensity level of the sound is

increased by 8 dB, find the new intensity of the sound.


Let the new intensity of the sound be .

Original intensity level = 10 log)(

)(dB

10 log)(

)(– 10 log

)(

)(= ( )

=


7. The intensity of the sound in a taxi is 10–6 W/m2.

(a) Find the intensity level of the sound.

(b) If the intensity level of the sound is halved now, find the new intensity of the sound.

Take out the common factor.

61



� Consolidation Exercise 8D

[In this exercise, unless otherwise stated, give the answers correct to 1 decimal place if necessary.]

Level 1

1. An earthquake of magnitude 9 occurred in an area. By using the formula M = log A + 4.24, find the

amplitude of the earthquake, correct to 3 significant figures.

2. An earthquake occurred in a region. The amplitude of the earthquake was measured to be 42 units. By

using the formula M = log A + 4.24, find the magnitude of the earthquake.

3. An earthquake occurred in two cities P and Q. The amplitudes recorded were 80 units and 320 units

respectively. Find the difference in the magnitudes of the earthquake in the two cities.

4. Two earthquakes occurred in a city. The amplitude recorded in the first earthquake was 5 times that

recorded in the second earthquake. Find the difference in the magnitudes of the two earthquakes.

5. The magnitude of an earthquake is 8. Using the formula M =0

log3

2

E

E, where M represents

the

magnitude of the earthquake, E units represents the energy released and E0 = 6.3 × 104, find the energy

released in the earthquake.

6. The intensity level of the sound recorded in a library is 30 dB. Find the intensity of the sound.

7. The intensity of the sound produced by an engine is 1 W/m2. Find the intensity level of the sound.

8. The intensity levels of sound measured in two offices are 35 dB and 36 dB respectively. Find the

difference in the intensities of sound of the two offices, correct to 3 significant figures.

9. People feel ear pain when hearing a sound at intensity level of 130 dB, which is called the threshold of

pain. The intensity of the sound recorded in a concert is 11 W/m2. Is the intensity level of the sound

recorded in the concert above the threshold of pain? Explain your answer.

10. For the two numbers 5234 and 2543, which one is larger? Explain your answer.

11. For the two numbers –0.4–321 and

123

8

1−

− , which one is larger? Explain your answer.

62

Level 2

12. The magnitude of an earthquake that occurred in area P was 8. Another earthquake occurred in area Q,

where the amplitude recorded was

50

1 of that recorded in area P. Find the magnitude of the

earthquake recorded in area Q.

13. An earthquake of magnitude 7.4 occurred in a town. Later, there was an aftershock of magnitude 7 in

the same town. How many times was the amplitude of the first earthquake as large as that of the

aftershock?

14. The magnitudes of two earthquakes that occurred in a region differ by 2. Find the ratio of the

amplitude of the stronger earthquake to that of the weaker earthquake.

15. An earthquake occurred in both city X and city Y. The amplitude recorded in city X was 300 units less

than that recorded in city Y. If the magnitudes of the earthquake in the two cities differ by 0.3, find the

amplitude recorded in city Y.

16. Three earthquakes occurred in a region. The magnitudes of the first and second earthquakes are 8.2

and 7.5 respectively. The amplitude of the third earthquake is

8

1 times that of the first earthquake.

(a) Which earthquake, the second or the third earthquake, has a larger magnitude? Explain your

answer.

(b) Find the difference between the magnitudes of the second and the third earthquakes.

17. The intensity of the sound from a speaker is 10–5 W/m2.

(a) Find the intensity level of the sound produced.

(b) If the intensity level of the sound is increased by 5 dB, find the new intensity of the sound,

correct to 3 significant figures.

18. The intensity level of the sound recorded near a residential area is 60 dB. After a construction site is

set up near the area, the intensity of the sound becomes 30 times of that before. Find the intensity

level of the sound after the construction site is set up.

19. The intensity level of a sound is 50 dB. If the intensity level of the sound is decreased by 20%, is the

new intensity of the sound

10

1 of the original? Explain your answer.

20. The intensity level of the sound recorded near a highway is 120 dB. If noise barrier A is set up, the

intensity of the sound will become

40

1 of that before. If noise barrier B is set up, the intensity level of

the sound will be decreased by 15 dB. Which noise barrier is more effective? Explain your answer.

21. If the intensity I W/m2 of a sound is increased to 100 times the original, the new intensity level of the

sound will be doubled.

(a) Find the original intensity of the sound.

(b) If the intensity I W/m2 of the sound is increased to 10 times the original, will the new intensity

level of the sound be increased by 50%? Explain your answer.

63

Answers

Consolidation Exercise 8D

1. 57 500 units 2. 5.9

3. 0.6 4. 0.7

5. 6.3 × 1016 units 6. 10–9 W/m2

7. 120 dB 8. 8.19 × 10–10

W/m2

9. yes 10. 5234

11. 123

8

1

− 12. 6.3

13. 2.5 times 14. 100 : 1

15. 601.4 units

16. (a) second earthquake

(b) 0.2

17. (a) 70 dB

(b) 3.16 × 10–5 W/m2

18. 74.8 dB

19. yes

20. noise barrier A

21. (a) 10–10 W/m2

(b) yes

64

F4B: Chapter 9A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )



___________ ( )



_________

65


Objective: To review the highest common factor (H.C.F.) and the least common multiple (L.C.M.) of

integers, factorization of polynomials, domain of a function and the factor theorem.

Highest Common Factor (H.C.F.) and Least Common Multiple (L.C.M.)

Find the H.C.F. and L.C.M. of each of the following groups of numbers. [Nos. 1–2]

1. 12, 16 2. 15, 21 �Review Ex: 1, 2

12 = ( )2 × ( ) 15 = ( ) × ( )

16 = ( )4 21 = ( ) × ( )

∴ H.C.F. = ∴ H.C.F. =

L.C.M. = L.C.M. =

Factorization

Factorize each of the following expressions. [Nos. 3–11] �Review Ex: 3–11

3. x2 – 3x + 2 4. x2 + 10x + 25 5. 3x2 + 2xy – y2

= = =

6. x2 – 4xy + 4y2 7. 4x2 – 9y2 8. 2x + 2 + xy + y

= = =

9. 4x2 – 9(x + 1)2 10. x3 + 64 11. 8x3 – 27y3

= = =

12. (a) Factorize a2 – 16. 13. (a) Factorize 9a2 + 24ab + 16b2.

(b) Hence, factorize a4 – 16. (b) Hence, factorize 9a2 + 24ab + 16b2 – 3a – 4b.

� � 64 = ( )3

�Review Ex: 12–14

a2 + 2ab + b2 ≡ (a + b)2

a2 – 2ab + b2 ≡ (a – b)2 a2 – b2 ≡ (a + b)(a – b)

a3 + b3 ≡ (a + b)(a2 – ab + b2) a3 – b3 ≡ (a – b)(a2 + ab + b2)

66

Domain of a Function

Find the domain of each of the following functions. [Nos. 14–15]

14. f(x) = x2 – 3x + 2 15. f(x) =7

2

−x �Review Ex: 15, 16

Domain: ∵ The value of the denominator

cannot be 0.

∴ ( ) ≠ 0

∴ The domain is

.

Factor Theorem

16. Let f(x) = x3 + 5x2 + 3x – 9. 17. Let f(x) = x3 + 4x2 + x – 6. �Review Ex: 17

(a) Show that x – 1 is a factor of f(x). (a) Show that x + 2 is a factor of f(x).

(b) Factorize f(x). (b) Factorize f(x).


18. It is given that x + 1 is a factor of f(x) = 2x3 – 7x2 + k.


(b) Factorize f(x).

67


Objective: To understand the concepts of H.C.F. and L.C.M. of polynomials.

Highest Common Factor (H.C.F.) and Least Common Multiple (L.C.M.)

(a) Among the common factors of two or more polynomials, if the polynomial P is the one having the

highest degree, then P is called the highest common factor of the polynomials.

(b) Among the common multiples of two or more polynomials, if the polynomial Q is the one having the

lowest degree, then Q is called the least common multiple of the polynomials.

Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 1–2]

1. x4, xy2 2. 6x5y, 9x3y2 � Ex 9A: 1, 2

x4 = ( ) 6x5y = ( ) × ( ) × ( ) × ( )

xy2 = ( ) × ( ) 9x3y2 = ( ) × ( ) × ( )

∴ H.C.F. =

L.C.M. =


Find the H.C.F. and L.C.M. of x3(x + 1) and

x(x + 1)2.

x3(x + 1) = x3 × (x + 1)

x(x + 1)2 = x × (x + 1)2 ∴ H.C.F. = x(x + 1)

L.C.M. = x3(x + 1)2

Find the H.C.F. and L.C.M. of x2(x − 3) and

x(x − 3)5.

x2(x − 3) =

x(x − 3)5 = ∴ H.C.F. =

L.C.M. =


3. x2(x + 3)3, x3(x + 3) 4. x5(3x – 1)3, x3(3x – 1)2 � Ex 9A: 3–10

x2(x + 3)3 =

x3(x + 3) =

∴ H.C.F. =

L.C.M. =

5. (x + 2)(x − 4)3, (x + 2)2(x − 4)2 6. (x − 5)6(2x + 1), (x − 5)3(2x + 1)4

(x + 2)(x − 4)3 =

(x + 2)2(x − 4)2 =

∴ H.C.F. =

L.C.M. =

�

�Take the highest power in each group with the same base.

Take the lowest power in each group with the same base.

x and x + 1 are two bases.

( ) and ( ) are two bases.

68

7. 2(x − 1)4(x − 8), 6(x − 1)2(x − 8)3 8. (x + 2)(3x − 4), (x − 7)(3x − 4)2


Find the H.C.F. and L.C.M. of x2 + 2x + 1 and

x2(x + 1).

x2 + 2x + 1 = (x + 1)2

x2(x + 1) = x2(x + 1) ∴ H.C.F. = x + 1

L.C.M. = x2(x + 1)2

Find the H.C.F. and L.C.M. of x2 − 3x + 2 and

x(x − 2)5.

x2 − 3x + 2 =

x(x − 2)5 = ∴ H.C.F. =

L.C.M. =


9. (x + 2)(x + 4)2, x2 + 5x + 4 10. x2 − 6x + 9, x2 + x − 12 � Ex 9A: 11–14

11. 2x2 − 3x + 1, 4x2 − 1 12. 2x2 + 6x + 4, x2 − 6x − 7


13. Let f(x) = 3xy3 − 6x2y2 and g(x) = xy2 – 5x2y + 6x3.

(a) Factorize f(x) and g(x).

(b) Find the H.C.F. and L.C.M. of f(x) and g(x).

69


9 Rational Functions


Level 1


1. (a) x4, xy4 (b) ab, a3b2

2. (a) uv2w4, u2v5w3 (b) p4q6r2, p7q3r9

3. (a) x2y3z, x2yz2, xy4z (b) abc2, a4bc, a3c

4. (a) 6p2q3, 4p3q2 (b) 8u3vw5, 9u7v8w

(c) 4y4z2, 10xy6z5 (d) 12a5b4, 15a3b2c, 5a4c3

Find the H.C.F. of each of the following groups of polynomials. [Nos. 5–7]

5. (a) x4(x – 3)2, x(x – 3)5

(b) x3(x + 5)(x – 2), x2(x + 5)4

6. (a) (x – 2)(x + 1)2(x + 4), (x + 1)(x + 4)(x – 2)3

(b) (x – 1)(x – 2)(2x + 3)3, (x – 1)(x – 2)2(2x + 3)2

7. (a) 9x2(x + 2)3, 3x3(x – 1)(x + 2)

(b) 8x4(x + 7)2(2x – 1)4, 20x(x + 7)3(2x – 1)2

Find the L.C.M. of each of the following groups of polynomials. [Nos. 8–10]

8. (a) x2(x – 3)2, x(x – 3)4

(b) (x – 9)3(x + 6)2, (x + 6)(x – 9)2

9. (a) x2(x – 1)(x – 4)2, x3(x – 1)2(x – 4)4

(b) (x + 2)(x – 4)2(3x – 2)3, (x – 4)(x + 2)2(3x – 2)

10. (a) 2(3x + 1)2(2x – 7)3, 6(3x + 1)(2x – 7)4

(b) 16x4(x – 5)3(x + 8)2, 12x3(x – 5)(x + 8)3


11. (x – 6)5(4x + 9)2, (x – 6)2(4x + 9)3, (x – 6)4(4x + 9)

12. x(x + 3)4(2x + 5)2, x2(x + 3)5, (x + 3)3(2x + 5)4

70

13. 5x2(x – 8)3(3x – 2), 10x(x – 8)(3x – 2)4, 15x4(x – 8)4(3x – 2)3

14. 9x(x + 1)(2x + 3)3, 15x2(x + 1)3(x + 2)5, 6x3(x + 2)2(2x + 3)4

15. (a) Factorize x2 + x – 2 and 2x2 + 4x.

(b) Find the H.C.F. of the two polynomials in (a).

16. (a) Factorize 6x2 – x – 1 and 8x2 – 8x + 2.

(b) Find the H.C.F. of the two polynomials in (a).

17. (a) Factorize x2 – 6x + 5 and 2x2 – 50.

(b) Find the L.C.M. of x2 – 6x + 5 and 2x2 – 50.

18. (a) Factorize 2x2 – 3x – 5 and 3x3 + 6x2 + 3x.

(b) Find the L.C.M. of 2x2 – 3x – 5 and 3x3 + 6x2 + 3x.

19. Let f(x) = x4 – 2x3 – 15x2 and g(x) = 27x – 3x3. Find the H.C.F. of f(x) and g(x).

20. Let h(x) = 98x3 – 28x2 + 2x and k(x) = 7x4 – 15x3 + 2x2. Find the H.C.F. of h(x) and k(x).

21. Let p(x) = 4x2 – 5x – 6 and q(x) = 12x3 + 9x2. Find the L.C.M. of p(x) and q(x).

22. Let u(x) = 3x2 + 14x – 5 and v(x) = 18x3 – 12x2 + 2x. Find the L.C.M. of u(x) and v(x).

23. Find the H.C.F. and L.C.M. of x2 – 8x + 16 and 5x2 – 10x – 40.

24. Find the H.C.F. and L.C.M. of x3 – 64x and x2 + 2x – 48.

25. Find the H.C.F. and L.C.M. of 4x2 – 4, 6x2 – 14x + 8 and 8x2 + 4x – 12.

26. Find the H.C.F. and L.C.M. of 3x2 + 7x – 20, 12x2 – 2x – 30 and 18x2 – 60x + 50.

Level 2


27. 2p2 – 18q2, 5p2 – 14pq – 3q2

28. 6a2 – 10ab – 4b2, 4a2 – 16b2

29. 6u2 – 21uv – 12v2, 12u2 + 12uv + 3v2

30. 2x3 + 3x2y – 2xy2, 2x3y + 2x3 – x2y2 – x2y

71

31. 45h2 – 5k2, 15h2 + 10hk – 5k2, 6h2 – 5hk + k2

32. 2m2 + 4mn + 2n2, 12m2 + 9mn – 3n2, m3 + n3

33. Let f(x) = x2 – 4x – 21 and g(x) = x3 + 2x2 – 5x – 6.

(a) Factorize f(x).

(b) Show that g(2) = 0. Hence, factorize g(x).

(c) Find the H.C.F. of f(x) and g(x).

34. Let p(x) = 3x2 + 20x + 12 and q(x) = 2x3 + 13x2 + 5x – 6.

(a) Factorize p(x).

(b) Show that 2x – 1 is a factor of q(x). Hence, factorize q(x).

(c) Find the L.C.M. of p(x) and q(x).

35. Let u(x) = x3 + 6x2 – 27x and v(x) = x3 – 2x2 – 15x + 36.

(a) Factorize u(x).

(b) Find the value of v(3). Hence, factorize v(x).

(c) Find the L.C.M. of u(x) and v(x).

36. Let h(x) = 9x3 + 30x2 + 24x and k(x) = 3x4 + 4x3 – 9x2 – 10x.

(a) Factorize h(x).

(b) Find the value of k(–2). Hence, factorize k(x).

(c) Find the H.C.F. of h(x) and k(x).

37. Let f(x) = x3 + kx2 + 8x – 4 and g(x) = x2 + 3x – 4, where k is a constant. It is given that x – 1 is a factor

of f(x).


(b) Find the H.C.F. and L.C.M. of f(x) and g(x).

38. Let u(x) = 4x3 + 2ax2 + 13x + b and v(x) = ax2 – 2x – b, where a and b are constants. It is given that the

H.C.F. of u(x) and v(x) is 2x + 1.

(a) Find the values of a and b.

(b) Find the L.C.M. of u(x) and v(x).

39. Let R = 2x3 + 3x2 – 1 and S = 2x3 + 7x2 + 2x – 3. Find the H.C.F. of R and S.

40. Let P = x3 + 2x2 – 15x and Q = x3 + 6x2 – 15x – 100. Find the H.C.F. of P and Q.

41. Let H = 9x3 + 15x2 – 6x and K = 3x3 + 11x2 + 8x – 4. Find the L.C.M. of H and K.

42. Let M = 4x4 – 24x3 + 36x2 and N = 8x3 – 30x2 + 16x + 6. Find the L.C.M. of M and N.

72

43. Let r(x) = 4x2 – 18x + 20 and s(x) = r(x) + 5x3 – 19x2 + 18x.

(a) Find s(x).

(b) Find the L.C.M. of r(x) and s(x).

44. Let f(x) = 15x2 + 5x – 20 and g(x) = 18x3 – 26x + 8 – x[f(x)].

(a) Find g(x).

(b) Find the H.C.F. of f(x) and g(x).

45. Let f(x) = x3 + 2x2 – 24x and g(x) = x3 – 27x + 54.

(a) Find the value of g(3). Hence, factorize g(x).

(b) Let P and Q be the H.C.F. and L.C.M. of f(x) and g(x) respectively.

(i) Find P and Q.

(ii) Are the results of f(x)g(x) and P × Q equal? Explain your answer.

46. Let R = x3 – x2y + xy2 – y3 and S = x3y – x3 + xy3 – xy2.

(a) Find the H.C.F. and L.C.M. of R and S.

(b) Using the results of (a), find R × S.

47. The H.C.F. and L.C.M. of two polynomials are ab4 and 4a3b5c2 respectively. If the first polynomial is

a3b4, find the second polynomial.

48. The H.C.F. and L.C.M. of three polynomials are uv and 8u3v2w6 respectively. If the first and the

second polynomials are u3vw4 and 4u2vw6 respectively, find the third polynomial.

49. Let f(x) = 6x2 + 13x – 28 and g(x) be a polynomial. If the H.C.F. and L.C.M. of f(x) and g(x) are 3x – 4

and 6x3 + 25x2 – 2x – 56 respectively, find g(x).

50. Let f(x) = x3 + kx – 30, where k is a constant. It is given that x + 2 is a factor of f(x).


(b) Factorize f(x).

(c) Let g(x) = x2 – 3x – 10 and h(x) be a polynomial. If the H.C.F. and L.C.M. of g(x) and h(x) are

x – 5 and f(x) respectively, find h(x).

73

Answers


1. (a) H.C.F. = x, L.C.M. = x4y4

(b) H.C.F. = ab, L.C.M. = a3b2

2. (a) H.C.F. = uv2w3, L.C.M. = u2v5w4

(b) H.C.F. = p4q3r2, L.C.M. = p7q6r9

3. (a) H.C.F. = xyz, L.C.M. = x2y4z2

(b) H.C.F. = ac, L.C.M. = a4bc2

4. (a) H.C.F. = 2p2q2, L.C.M. = 12p3q3

(b) H.C.F. = u3vw, L.C.M. = 72u7v8w5

(c) H.C.F. = 2y4z2, L.C.M. = 20xy6z5

(d) H.C.F. = a3, L.C.M. = 60a5b4c3

5. (a) x(x – 3)2 (b) x2(x + 5)

6. (a) (x – 2)(x + 1)(x + 4)

(b) (x – 1)(x – 2)(2x + 3)2

7. (a) 3x2(x + 2)

(b) 4x(x + 7)2(2x – 1)2

8. (a) x2(x – 3)4

(b) (x – 9)3(x + 6)2

9. (a) x3(x – 1)2(x – 4)4

(b) (x + 2)2(x – 4)2(3x – 2)3

10. (a) 6(3x + 1)2(2x – 7)4

(b) 48x4(x – 5)3(x + 8)3

11. H.C.F. = (x – 6)2(4x + 9),

L.C.M. = (x – 6)5(4x + 9)3

12. H.C.F. = (x + 3)3, L.C.M. = x2(x + 3)5(2x +

5)4

13. H.C.F. = 5x(x – 8)(3x – 2),

L.C.M. = 30x4(x – 8)4(3x – 2)4

14. H.C.F. = 3x,

L.C.M. = 90x3(x + 1)3(x + 2)5(2x + 3)4

15. (a) x2 + x – 2 = (x – 1)(x + 2),

2x2 + 4x = 2x(x + 2)

(b) x + 2

16. (a) 6x2 – x – 1 = (2x – 1)(3x + 1),

8x2 – 8x + 2 = 2(2x – 1)2

(b) 2x – 1

17. (a) x2 – 6x + 5 = (x – 1)(x – 5),

2x2 – 50 = 2(x + 5)(x – 5)

(b) 2(x – 1)(x + 5)(x – 5)

18. (a) 2x2 – 3x – 5 = (x + 1)(2x – 5),

3x3 + 6x2 + 3x = 3x(x + 1)2

(b) 3x(x + 1)2(2x – 5)

19. x(x + 3)

20. x(7x – 1)

21. 3x2(x – 2)(4x + 3)

22. 2x(x + 5)(3x – 1)2

23. H.C.F. = x – 4, L.C.M. = 5(x + 2)(x – 4)2

24. H.C.F. = x + 8, L.C.M. = x(x – 6)(x + 8)(x –

8)

25. H.C.F. = 2(x – 1),

L.C.M. = 4(x – 1)(x + 1)(2x + 3)(3x – 4)

26. H.C.F. = 3x – 5,

L.C.M. = 2(x + 4)(2x + 3)(3x – 5)2

27. H.C.F. = p – 3q,

L.C.M. = 2(p – 3q)(p + 3q)(5p + q)

28. H.C.F. = 2(a – 2b),

L.C.M. = 4(a – 2b)(a + 2b)(3a + b)

29. H.C.F. = 3(2u + v),

L.C.M. = 3(2u + v)2(u – 4v)

30. H.C.F. = x(2x – y),

L.C.M. = x2(y + 1)(2x – y)(x + 2y)

31. H.C.F. = 3h – k,

L.C.M. = 5(3h – k)(3h + k)(2h – k)(h + k)

32. H.C.F. = m + n,

L.C.M. = 6(m + n)2(4m – n)(m2 – mn + n2)

33. (a) (x + 3)(x – 7)

(b) (x + 1)(x – 2)(x + 3)

(c) x + 3

34. (a) (x + 6)(3x + 2)

(b) (x + 1)(x + 6)(2x – 1)

(c) (x + 1)(x + 6)(2x – 1)(3x + 2)

35. (a) x(x – 3)(x + 9)

(b) 0, (x – 3)2(x + 4)

(c) x(x – 3)2(x + 4)(x + 9)

36. (a) 3x(x + 2)(3x + 4)

(b) 0, x(x + 1)(x + 2)(3x – 5)

(c) x(x + 2)

74

37. (a) –5

(b) H.C.F. = x – 1,

L.C.M. = (x + 4)(x – 1)(x – 2)2

38. (a) a = 8, b = 3

(b) (x + 3)(2x + 1)2(4x – 3)

39. (x + 1)(2x – 1)

40. x + 5

41. 3x(x + 2)2(3x – 1)

42. 4x2(x – 1)(x – 3)2(4x + 1)

43. (a) 5x3 – 15x2 + 20

(b) 10(x + 1)(x – 2)2(2x – 5)

44. (a) 3x3 – 5x2 – 6x + 8

(b) (x – 1)(3x + 4)

45. (a) 0, (x – 3)2(x + 6)

(b) (i) P = x + 6, Q = x(x – 3)2(x – 4)(x +

6)

(ii) yes

46. (a) H.C.F. = x2 + y2,

L.C.M. = x(y – 1)(x – y)(x2 + y2)

(b) x(y – 1)(x – y)(x2 + y2)2

47. 4ab5c2

48. 8uv2

49. (x + 2)(3x – 4)

50. (a) –19

(b) (x + 2)(x + 3)(x – 5)

(c) (x + 3)(x – 5)

75

F4B: Chapter 9B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)



(Full Solution)




___________ ( )


○ Complete and Checked ○ Problems encountered


___________

76

○ Skipped ( )



___________ ( )



_________

77


Objective: To know the definition of rational functions.

Definition of Rational Functions

If a function can be expressed in the form

Q

P (where P and Q are polynomials, and Q ≠ 0),

then the function is called a rational function.


Find the domain of each of the following rational

functions.

(a) f(x) =6

3

+

−

x

x (b) g(x) =

2)1(

2

−x

(a) ∵ The value of the denominator x + 6 cannot

be 0.

∴ x + 6 ≠ 0

i.e. x ≠ −6

∴ The domain is all real numbers except −6.

(b) ∵ The value of the denominator (x − 1)2

cannot be 0.

∴ (x − 1)2 ≠ 0

x − 1 ≠ 0

i.e. x ≠ 1

∴ The domain is all real numbers except 1.

Find the domain of each of the following rational

functions.

(a) f(x) =5

3

−

+

x

x (b) g(x) =

2)2(

9

+x

(a) ∵ The value of the denominator ( )

cannot be ( ).

∴ ( ) ≠ 0

i.e. x ≠ ( )

∴

(b) ∵ The value of the denominator ( )

cannot be ( ).

∴ ( ) ≠ 0

i.e. x ≠ ( )

∴

Find the domain of each of the following rational functions. [Nos. 1–2]

1. f(x) =9

73

+

+

x

x 2. g(x) =

2)5(

14

−

−

x

x � Ex 9B: 1, 2

∵ The value of the denominator ( )

cannot be ( ).

∴

�

x

1is undefined when x = 0.

78


Simplify

xx

x

32

962 +

+.

xx

x

32

962 +

+=

)32(

)32(3

+

+

xx

x

=x

3

Simplify

xx

x

88

552 +

+.

xx

x

88

552 +

+=

Simplify each of the following rational functions. [Nos. 3–12]

3. xx

x

2

32

2

+= 4.

xx

xx

66

122

2

+

++= � Ex 9B: 3–8

5. xx

x

123

162

2

−

−= 6.

9

962

2

−

++

x

xx=

7. 169

192

2

+−

−

xx

x= 8.

168

122

2

+−

−−

xx

xx=

9. 572

2542

2

++

−

xx

x= 10.

32

1272

2

−−

+−

xx

xx=

11. 43

232

2

−+

−−

xx

xx= 12.

7103

1342

2

++

−+

xx

xx=


13. (a) Find the domain of the rational function f(x) =2)4(

53

+

+

x

x.

(b) Give another example of a rational function which has the same domain as f(x).

�Take out the common factor 3.

�Take out the common factor x.

79


Objective: To perform the multiplication and division of rational functions.

Multiplication and Division of Rational Functions

Step 1: Factorize the numerator and the denominator of each rational function in an expression.

Step 2: Reduce the fraction.

Step 3: Multiply the rational functions after reducing the fraction.


Simplify

13 +x

x×

xx

x

7

132 +

+.

13 +x

x×

xx

x

7

132 +

+

=13 +x

x×

)7(

13

+

+

xx

x

=7

1

+x

Simplify

xx

x

+

−−2

5×

5+x

x.

xx

x

+

−−2

5×

5+x

x

=) (

) (×

) (

) (

=

Simplify each of the following expressions. [Nos. 1–6]

1. 3

2

5

)4(

4

4

x

x

x

x +×

+ 2.

x

x

x

x

9

)2(

2

3 23−

×

− � Ex 9B: 9–15

=) (

) ( =

) (

) (×

) (

) (

=

3. )2)(1(

66

1

)1(

−−

+×

+

−

xx

x

x

xx 4.

52

9

+x×

xx

xx

3

522

2

+

+

=) (

) (×

) (

) ( =

=

5. xx

x

34

42 +

−×

xx

x

287

82

2

− 6.

)3)(2(

93 2

++

+

xx

xx×

2

42

−

−

x

x

=

=

�

� Reduce the fraction after factorizing the numerator and the denominator.

80


Simplify

2

23

−

+

x

x÷

x

x 69 +.

2

23

−

+

x

x÷

x

x 69 +=

2

23

−

+

x

x×

69 +x

x

=2

23

−

+

x

x×

)23(3 +x

x

=)2(3 −x

x

Simplify

x

x 88 −÷

3

22

+

−

x

x.

x

x 88 −÷

3

22

+

−

x

x=

) (

) (×

) (

) (

=


7. 65

2

+

−

x

x÷

x

xx

7

22 − 8.

2

2

4

5

x

xx −÷

9

153

+

−

x

x � Ex 9B: 16–20

=) (

) (×

) (

) ( =

) (

) (×

) (

) (

=

=

9. xx

x

66

242 +

−÷

x

xx

9

)3)(12( +− 10.

)4)(52(

322

++

−−

xx

xx÷

xx

x

123

32 +

−

=) (

) (×

) (

) ( =

=

11. 155

)5)(2(

+

−+

x

xx÷

9

1022 −

−

x

x 12.

2

34

44

33 2

2 +

−−÷

++

−

x

xx

xx

x

=

=


13. Simplify424

)1)(2( 22

+

−÷

+×

+

−+

x

xx

x

x

x

xx.

� Convert the operation

from ÷ to ×.

81




Level 1


1. 5

2

−x 2.

x

x

−

+

4

2

3. 2)3(

1

+x 4.

9

32 +

−

x

x


5. 93

6

+x 6.

xx

x

168

42

3

+

7. )3(2

124

−

−

xx

x 8.

xx

x

2

1052 +

+

9. 2425

52

x

x

−

− 10.

98

12 −−

+

xx

x

11. 12

932 −+

−

xx

x 12.

2

2

)2(

82

−

−+

x

xx

13. 12

)5)(3(2 −−

−+

xx

xx 14.

992

3442

2

++

−+

xx

xx

15. 32162

24102

2

+−

+−

xx

xx 16.

5656

6492

2

−+

−

xx

x


17. 32

1

)1(

2

x

x

x

x +×

+ 18.

3

2

)3(

8

4

3

−×

−

h

h

h

h

19. 2

)5)(4(

5

22

xx

xx

x

x

+

++×

+

+ 20.

)6(

9

7

62

23

−

−×

+

−

kk

k

kk

kk

82

21. xx

x

x

xx

23

123

82

482

2

−

+×

+

− 22.

pp

p

pp

pp

+

−×

−

+22

2

3

186

84

26

23. )54)(1(

3

62

12 22

−−

−×

−

+−

xx

xx

x

xx 24.

164

2

187

82 2

2

23

−

−+×

−−

−

t

tt

tt

tt

25. 24

3

)2()2( +÷

+ x

x

x

x 26.

)1)(5(

7

1

)5(

−+÷

−

+

uu

u

u

uu

27. )5(

12

6

22

2

+

+÷

+

+

xx

x

xx

xx 28.

6

)12(3

9

22

23

+

−÷

−

−

v

vv

vv

vv

29. xx

x

x

x

−

+÷

−

+22 3

7

19

26 30.

ss

s

s

ss

62

4

155

1262

22

+

−÷

+

−

31. 96

4

)2)(3(

822

223

++

−÷

−+

−

xx

xx

xx

xx 32.

34

1

68

32 2

2

2

+

−÷

+

−−

z

z

zz

zz

Level 2


33. 6)1(

742 ++

−

x

x 34.

xx

x

5

382 −

+

35. 124

72 −−

+

xx

x 36.

2

2

)8(9

53

−−

++

x

xx


37. 2

4

936

464

x

x

−

− 38.

27

1523

2

−

−−

x

xx

39. 22

2

16

520

yx

yxy

−

− 40.

22

2

45

123

nmnm

mnm

+−

−

41. 22

22

4

812

qp

qpqp

−

++ 42.

22

22

2

65

khkh

khkh

+−

−+

43. 22

22

49

2110

ba

baba

−

+− 44.

22

33

82

8

vuvu

vu

−−

+

83


45. 81

168

127

9102

2

2

2

−

+−×

+−

+−

x

xx

xx

xx

46. 65

82

96

122

2

2

2

++

−−÷

++

−−

xx

xx

xx

xx

47. 34

183

32283

642

2

2

2

+−

−+×

+−

−

xx

xx

xx

x

48. 2510

145

152

1872

2

2

2

+−

−+÷

−−

−+

xx

xx

xx

xx

49. 22

22

36

209

305

5

sr

srsr

sr

sr

−

+−÷

+

−

50. 3322

22

27

424

3612

183

cb

bc

cbcb

cbcb

+

−×

+−

−−

51. )1(3

32223

−×−

−÷

−a

aa

a

aa

a

52. )2)(1(

93

14

32 −+

+÷

+×

−

+

xx

x

x

x

x

x

53. 1)3(

2

2

3 3

2

22

−

−÷

+

−−×

−

+

t

tt

t

tt

t

tt

54. 12

1

44

2

84

2232 +

×++

−÷

+

+

wwww

w

ww

w

55. kh

kh

khkh

khkhkh

+

−÷

++

++×+

33

22

22 8

34

42)3(

56. zy

zy

zy

yzy

zyzy

yzy

5

325

352

25 2

22

23

+

+×

−

−÷

++

−

84

Answers

Consolidation Exercise 9B

1. all real numbers except 5

2. all real numbers except 4

3. all real numbers except –3

4. all real numbers

5. 3

2

+x

6. )2(2

2

+x

x

7. x

2

8. x

5

9. 52

1

+−

x

10. 9

1

−x

11. 4

3

+x

12. 2

4

−

+

x

x

13. 4

5

−

−

x

x

14. 3

12

+

−

x

x

15. )4(2

6

−

−

x

x

16. 72

83

+

+

x

x

17. )1(

22 +xx

18. )3(2

1

−hh

19. x

x 4+

20. 7

9

+

−

k

k

21. 23

)12(6

−

−

x

x

22. )2(

)3(3

−

−

pp

p

23. )54(2

)1(

−

−

x

xx

24. )9(2

)1(2

−

−

t

tt

25. 2

2

)2( +x

x

26. 7

)5( 2+u

27. 6

)5(

+

+

x

xx

28. )9(3

6

−

+

v

v

29. 7

2

+x

x

30. )2(5

12 2

+s

s

31. 2

)3(2

−

+

x

xx

32. )1(2

3

−

−

zz

z

33. all real numbers

34. all real numbers except 0 and 5

35. all real numbers except –2 and 6

36. all real numbers except 5 and 11

37. 9

)4(4 2 +x

38. 93

522 ++

+

xx

x

39. yx

y

+4

5

40. nm

m

−

3

41. qp

qp

−

+

2

6

42. kh

kh

−

+ 6

43. ba

ba

7

3

+

−

44. vu

vuvu

4

42 22

−

+−

45. )9)(3(

)4)(1(

+−

−−

xx

xx

46. 1

47. )43)(1(

)8)(6(

−−

++

xx

xx

48. )7)(3(

)9)(5(

++

+−

xx

xx

85

49. )4(5

6

sr

sr

−

−

50. 22 93

4

cbcb +−−

51. 2

52. )2(3 +x

x

53. 3

1

+t

54. )2(4

12

−

+

w

w

55. kh 2

1

−

56. zy

zy

+

−

86

F4B: Chapter 9C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)



(Full Solution)

87




___________ ( )




___________ ( )

Maths Corner Exercise 9C Multiple Choice


___________ ( )



_________

88


Objective: To perform the addition and subtraction of rational functions.

Addition and Subtraction of Rational Functions

Step 1: Simplify each rational function in an expression.

Step 2: Find the L.C.M. of the denominators of the rational functions.

Step 3: Take the L.C.M. as the denominator of each rational function.

Step 4: Perform addition or subtraction of the new numerators.

Step 5: Simplify the results obtained.


Simplify1

1

3

2

++

+ xx.

1

1

3

2

++

+ xx

=)1)(3(

)3()1(2

++

+++

xx

xx

=)1)(3(

322

++

+++

xx

xx

=)1)(3(

53

++

+

xx

x

Simplify3

2

2

1

−+

+ xx.

3

2

2

1

−+

+ xx

=) )( (

) (2) ( +

=) )( (

) (

=


1. 5

4

2

3

++

− xx 2.

4

1

)4(

4

++

+ xxx �Ex 9C: 3, 4, 7, 9, 11

= =

3. 3

1

62

12 −

+− xxx

4. 5

1

25

152 +

+−

+

xx

x

=3

1

)3)( (

1

−+

− xx =

=

�

⊳ The L.C.M. of the two denominators is (x + 3)(x + 1).

In the final step, take out the common factor

in the numerator.

Remember to simplify the results.

a2 – b2 ≡ (a + b)(a – b)

Add the two polynomials in the numerator and simplify the result.

L.C.M. of denominators = ( )( )

89


Simplify5

2

6

3

−−

− xx.

5

2

6

3

−−

− xx=

)5)(6(

)6(2)5(3

−−

−−−

xx

xx

=)5)(6(

122153

−−

+−−

xx

xx

=)5)(6(

3

−−

−

xx

x

Simplifyxx

1

1

4−

−.

xx

1

1

4−

−=

) )( (

) () (4 −

=) )( (

) (

=) )( (

) (


5. 4

1

52

2

−−

+ xx 6.

1

3

)1)(1(

4

−−

−+ xxx

x �Ex 9C: 5, 6, 8, 10, 12

= =

7. 3

2

6

32 +

−−+ xxx

x 8.

6

1

65

522 −

−−−

−

xxx

x

=3

2

) )( (

3

+−

x

x =

=


9. Simplify each of the following expressions.

(a) xxxx

3

)2(

2

2

4−

++

+ (b)

xxxxx

x 1

5

3

)5(

72

+−

−−

90


Objective: To perform mixed arithmetic operations of rational functions.

Mixed Arithmetic Operations of Rational Functions

The mixed arithmetic operations of rational functions follow the principle of ‘multiplication and division

first, then addition and subtraction’.


Simplify2

11

2

2

−×

−+

− xx

x

x.

2

11

2

2

−×

−+

− xx

x

x

=)2(

1

2

2

−

−+

− xx

x

x

=)2(

)1(2

−

−+

xx

xx

=)2(

13

−

−

xx

x

Simplify54

3

11

1

−×

+−

+ xx

x

x.

54

3

11

1

−×

+−

+ xx

x

x

=) )( (

) (

1

1−

+x

=) )( (

) () ( −

=) )( (

) (


1. 12

2

1

1

12

1

+

+×

−+

+ x

x

xx 2.

3

3

3

2

9

22 −

×+

−− xxx

x �Ex 9C: 23

= =) )( (

6

) )( (

2−

x

=

3. 12

1

592

1352 −

−−+

−×

+

xxx

x

x

x 4.

x

x

xxx

x 1

4

1

45

142

−×

−+

+−

+

= =

�

⊳ Do ‘×’ first.

91


Simplify11

45

−÷

−−

x

x

xx.

11

45

−÷

−−

x

x

xx=

x

x

xx

1

1

45 −×

−−

=xx

45−

=x

1

Simplifyx

x

xx

23

2

2 +÷+

+.

x

x

xx

23

2

2 +÷+

+=

) (

) (3

2

2×+

+ xx

=) (

) (

2

2+

+x

=) (

) (


5. x

x

xx 3

4

1

2

4

6 +÷

+−

+ 6.

x

x

xx −÷

−−

+ 55

1

3

4 �Ex 9C: 24

= = ) (

) (

5

1

3

4 −×

−−

+ xx

=

7.

−+÷

+

−

4

21

23

22

xx

xx 8. 1

1

1−

−÷

+ x

x

x

x

= =


9. Simplify

xx

x

1

1

−

+.

Convert

÷ to ×.

92



� Consolidation Exercise 9C

Level 1


1. 5

7

5

3

−+

− xx 2.

6

7

6

32

+

−+

+

+

x

x

x

x

3. 4

2

4

6

+−

+ xx 4.

9

2

9

45

−

+−

−

−

x

x

x

x

5. 3

4

2

4

++

+ xx 6.

23

1

52

3

++

+ xx

7. 5

3

12

3

+−

+ xx 8.

25

3

13

2

−−

− xx

9. )4(

1

)1(

1

++

− xxxx 10.

)2)(3(

3

)2)(2(

2

+−−

+− xxxx

11. xx

x

−+

− 8

64

8

2

12. 2

2

2 53

37

35

89

xx

xx

xx

x

+−

−−

+−

−

13. )3)(5(

3

5

2

−++

+ xxx 14.

)25)(14(

7

14

5

+−−

− xxx

15. 9

3

)4(

8222 −

++

−

−

x

x

x

x 16.

)9)(7(

213

482

1222 +−

−−

−+

−

xx

x

xx

x

17. 43

4

43

52 +

+−+ xxx

18. 2

5

2115 2 −−

+− xxx

x

19. 4129

7

23

42 +−

+− xxx

20. 4

2

162 +−

− xx

x

21. 86

5

43

222 ++

+−+ xxxx

22. 54

3

152

422 −+

−−+ xxxx

23. xxx

5

1

2

1

3−

++

− 24.

4

2

5

3

3

1

−+

−−

− xxx

25. 4

3

9

8

4

2

−×

+

−+

− xx

x

x 26.

3

5

6

1

5

2

+

+÷

−−

+ x

x

xx

27. 5

34

5

7

−

−×−

− x

x

xx 28.

1

9

2

3

9

4

−

−÷

++

− x

x

xx

93

Level 2


29.

4

51

4

−+

x

30. 7

22

5−

+y

31. 222

22

yx

yx

yxy

y

−

−+

− 32.

2233

2

yxyx

yx

yx

x

++

+−

−

33. )92)(4(

73

92

4

4

4

−−

−−

−+

− xx

x

xx 34.

1

1

)1)(6(6

3

+−

+−−

− xxx

x

x

35. 1

1

1

2

1

223 +−

−+

+

−−

+ xx

x

x

x

x 36.

1

2

1

1

1

142 −

++

++

+

xxx

x

37. 22

11

xy

y

yx

x

xy

yx −+

−−

− 38.

22 )(

21

yx

yx

xyx

y

yx −

+−

−+

−

39. 222 )(

12

yx

yx

yxyx

x

+

−+

++

− 40.

33

22

2

2

yx

yxyx

xyx

yx

yx +

+−−

−

+−

−

41. 25

4

2

5

5

122

2

−

+×

−

−+

+ x

x

xx

xx

x 42.

−−

−÷

+−

−−

xxx

x

xx

xx

62

8

1242110

28322

2

43. If x =u

u1

+ and y =u

u1

− , express

xy

x

x

y 22 −− in terms of u. Give the answer in the simplest form.

44. If x =nm

mn

+ and y =

nm

mn

−, express

++

yxyx

11)( in terms of m and n. Give the answer in the

simplest form.

45. (a) Show that x + 1 is a factor of x3 – 9x2 + 14x + 24.

(b) Factorize x3 – 9x2 + 14x + 24.

(c) Simplify

24149

168

4

21

23

2

++−

+−×

−−

xxx

xx

x.

46. (a) Let k be a constant. Find the value of k such that x + 2 is a factor of x3 – 5x2 + kx + 12.

(b) Simplify

30

2416102

5

1

2

32

232

−−

+−−÷

+

−+

+

−

xx

xxx

x

x

x

x.

47. Consider the quadratic function z = −28 + 10x − x2.

(a) Find the maximum value of z and the corresponding value of x.

(b) Simplify

44

42

4

1

2

82

2

22

3

++

++÷

−×

+

−

zz

zz

zzz

z and express the answer in terms of z.

(c) Using the results of (a) and (b), find the minimum value of

44

42

4

1

2

82

2

22

3

++

++÷

−×

+

−

zz

zz

zzz

z.

94

Answers

Consolidation Exercise 9C

1. 5

10

−x

2. 6

43

+

−

x

x

3. 4

4

+x

4. 9

)3(2

−

−

x

x

5. )3)(2(

)52(4

++

+

xx

x

6. )23)(52(

)1(11

++

+

xx

x

7. )5)(12(

)4(3

++

−

xx

x

8. )25)(13(

1

−−

−

xx

x

9. )4)(1(

32

+−

+

xxx

x

10. )3)(2)(2( −+−

−xxx

x

11. x + 8

12. 3

13. )3)(5(

32

−+

−

xx

x

14. )25)(14(

325

+−

+

xx

x

15. )3)(4(

103

−−

−

xx

x

16. )9)(8(

6

++

+−

xx

x

17. )43)(1(

14

+−

+

xx

x

18. )15)(2(

245

−−

−

xx

x

19. 2)23(

112

−

−

x

x

20. )4)(4(

8

−+

−

xx

x

21. )4)(2)(1(

17

++−

−

xxx

x

22. )3)(1(

1

−− xx

23. )1)(1(

5

−+

+

xxx

x

24. )5)(4)(3(

)27(2

−−−

−

xxx

x

25. )9)(4(

65

+−

−

xx

x

26. )6)(5(

15

−+

−

xx

x

27. )5(

)4(3

−

+

xx

x

28. )9)(2(

57

−+

+

xx

x

29. 1

)4(4

+

−

x

x

30. )2(7

21

+

−

y

y

31. ))((

3

yxyx

x

−+

32. ))(( 22

2

yxyxyx

y

++−

33. )92)(4(

)5(9

−−

−

xx

x

34. )1)(6(

9

+−

+

xx

x

35. 1

3

+x

36. )1)(1)(1(

22

3

+−+ xxx

x

37. 22

)1)((

yx

xyyx −−

38. 2

2

)(

2

yxx

y

−−

39. 2

2

))((

4

yxyx

x

+−

40. )( yxx

y

+

41. )5)(2(

6

+−−

xx

42. 4

4

−x

x

43. )1)(1)(1(

22

2

+−+−

uuu

u

95

44. ))((

4 2

nmnm

m

−+

45. (b) (x + 1)(x – 4)(x – 6)

(c) 1

1

+x

46. (a) –8

(b) )2(2

53

+

−

x

x

47. (a) maximum value: –3, x = 5

(b) z

1 (c)

3

1−

Documents

Chapter 7 Exponential Functions 7A p.2 Chapter 8