Class 9 Maths Logarithm Notes

Class 9 Maths Logarithm Notes

Grade 9 Mathematics — Advanced Level Notes

1. Logarithm

If

bˣ = a

then

log_b a = x

Thus logarithms answer the question:

“To what power must the base be raised to produce the given number?”

Example:

2³ = 8

therefore

log₂ 8 = 3

2. Conditions for a Logarithm

For log_b a to be defined:

  • a > 0
  • b > 0
  • b ≠ 1

Therefore, the logarithm of zero or a negative number is not defined in this framework.

3. Important Basic Results

log_b 1 = 0

because b⁰ = 1.

log_b b = 1

because b¹ = b.

4. Main Laws of Logarithms

For positive M and N:

Product rule

log_b(MN) = log_b M + log_b N

Quotient rule

log_b(M/N) = log_b M − log_b N

Power rule

log_b(Mᵏ) = k log_b M

These rules convert multiplication into addition, division into subtraction, and powers into multiplication.

5. Change of Base

A logarithm can be expressed using another base:

log_a M = log_b M / log_b a

This is particularly useful when the required base is not directly available.

6. Solving Logarithmic Equations

A reliable method is:

  1. Check the domain conditions.
  2. Combine logarithms using the logarithm laws.
  3. Convert the resulting logarithmic equation into exponential form.
  4. Solve algebraically.
  5. Substitute the answer back into the original equation.

The last step is important because algebraic manipulation can sometimes produce an invalid value for a logarithm.

7. Graphical Idea

The chapter notes that logarithmic and exponential functions are inverse functions. Their graphs are reflections of one another in the line:

y = x

8. Applications

The text highlights logarithmic applications in:

  • earthquake measurement
  • pH/acidity
  • sound intensity
  • population growth
  • sports science

Exam Focus

Memorise the conditions + four main laws, but more importantly practise recognising which law to use.

Chapter 2 — Logarithms: Question Bank

A. Multiple Choice Questions (MCQs)

1. If 2⁵ = 32, then log₂ 32 is:

A. 2
B. 5
C. 16
D. 32

2. The value of log₁₀ 1000 is:

A. 2
B. 3
C. 10
D. 100

3. Which expression is equivalent to log_b 1?

A. 0
B. 1
C. b
D. −1

4. For log_b x to be defined in the real-number setting discussed in the chapter, which condition is necessary?

A. x < 0
B. b = 1
C. x > 0
D. b < 0

5. Which expression represents the logarithm of a product?

A. log_b(MN) = log_b M − log_b N
B. log_b(MN) = log_b M + log_b N
C. log_b(MN) = log_b M × log_b N
D. log_b(MN) = log_b(M + N)

6. log₃ 81 is equal to:

A. 3
B. 4
C. 9
D. 27

7. Which logarithmic expression is equivalent to log_b(M/N)?

A. log_b M + log_b N
B. log_b M − log_b N
C. log_b M × log_b N
D. log_b(M − N)

8. log₅ 5 equals:

A. 0
B. 1
C. 5
D. 25

9. Which law allows a power inside a logarithm to be written as a coefficient?

A. Product law
B. Quotient law
C. Power law
D. Addition law

10. If 10ˣ = 100, then x is:

A. 1
B. 2
C. 10
D. 100

11. The expression logₐ x asks:

A. What is x divided by a?
B. What power of a gives x?
C. What is a multiplied by x?
D. What is x raised to a?

12. Which of the following is not a valid base for a real logarithm?

A. 2
B. 10
C. 1
D. 5

B. Fill in the Blanks

13. A logarithm is the inverse operation of __________.

14. If bˣ = y, then log_b y = __________.

15. log_b 1 = __________.

16. log_b b = __________.

17. In log_b x, b is called the __________.

18. The argument of a real logarithm must be __________.

19. The base of a real logarithm must be positive and cannot be equal to __________.

20. log_b(MN) = log_b M + __________.

21. log_b(M/N) = log_b M − __________.

22. log_b(Mⁿ) = __________ log_b M.

23. The logarithm with base 10 is commonly called the __________ logarithm.

24. Logarithms can transform multiplication into __________.

C. True or False

25. log₂ 8 = 3.

26. log₁₀ 1 = 1.

27. log₅ 5 = 1.

28. The argument of a real logarithm may be zero.

29. The base of a logarithm may be 1.

30. log_b(MN) = log_b M + log_b N.

31. log_b(M/N) = log_b M + log_b N.

32. Logarithms and exponential functions are inverse operations.

33. log₃ 27 = 4.

34. A logarithmic equation should be checked against the original restrictions after solving.

D. Match the Following

Column AColumn B
35. log_b 1a. 1
36. log_b bb. Power of b
37. log_b(MN)c. 0
38. log_b(M/N)d. log_b M + log_b N
39. log_b(Mⁿ)e. log_b M − log_b N
40. Logarithmf. n log_b M

E. Evaluate the Logarithms

41. Evaluate log₂ 16.

42. Evaluate log₃ 81.

43. Evaluate log₅ 125.

44. Evaluate log₁₀ 10,000.

45. Evaluate log₇ 1.

46. Evaluate log₉ 9.

47. Find x if log₂ x = 6.

48. Find x if log₅ x = 3.

F. Use the Laws of Logarithms

49. Expand:

log₂(8 × 4)

using the product law.

50. Expand:

log₃(81/9)

using the quotient law.

51. Express:

log₅(x³)

in terms of log₅ x.

52. Expand:

log₂(4x).

53. Expand:

log₃(x²/y).

54. Condense:

log₄ x + log₄ y.

55. Condense:

log₇ a − log₇ b.

56. Condense:

3 log₂ x.

G. Convert Between Exponential and Logarithmic Forms

57. Convert 2⁴ = 16 into logarithmic form.

58. Convert 3⁵ = 243 into logarithmic form.

59. Convert log₄ 64 = 3 into exponential form.

60. Convert log₁₀ 100 = 2 into exponential form.

61. Write an equivalent logarithmic statement for 7² = 49.

62. Write an equivalent exponential statement for log₅ 625 = 4.

H. Solve the Logarithmic Equations

63. Solve:

log₂ x = 4

64. Solve:

log₃ x = 2

65. Solve:

log₅(x) = 3

66. Solve:

log₂(x) + log₂(4) = 5

67. Solve:

log₃(x) − log₃(3) = 2

68. Solve:

2 log₅ x = 4

69. Solve:

log₂(x²) = 6, subject to the appropriate domain restrictions.

70. Solve:

log₁₀ x + log₁₀ 10 = 3.

I. Conceptual / Reasoning Questions

71. Explain why logarithms are considered the inverse of exponentiation.

72. Why must the argument of a real logarithm be positive?

73. Why is log_b 1 = 0 for every valid base b?

74. Why is log_b b = 1?

75. Explain how the product law of logarithms changes multiplication into addition.

76. Explain the difference between a logarithm’s base and its argument.

77. A student writes log₂(4 + 8) = log₂4 + log₂8. Is this valid? Explain.

78. Why is it important to check the answer obtained from a logarithmic equation in the original equation?

J. Change of Base

79. Write log₂ 7 using logarithms with base 10.

80. Write log₅ 12 using natural logarithms.

81. State the change-of-base formula.

82. Explain one practical reason for using the change-of-base formula.

K. Application-Based Questions

83. A quantity is represented by 10³. Express its value and its logarithm to base 10.

84. A sound measurement uses the expression log₁₀ 1000. What value does this expression produce?

85. A quantity increases from 10² to 10⁵. How much does its base-10 logarithmic value increase?

86. Explain why logarithmic scales are useful when quantities vary over very large ranges.

87. Give two real-world fields in which logarithms are used and briefly explain what they help measure.

88. A measurement has a logarithmic value of 4 with base 10. What original quantity corresponds to this value?

L. Assertion–Reason Questions

For each question, choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

89. Assertion: log₂ 8 = 3.
Reason: 2³ = 8.

90. Assertion: log₅ 1 = 0.
Reason: Every valid non-zero base raised to the power zero gives 1.

91. Assertion: log_b(MN) = log_b M + log_b N.
Reason: Logarithms convert multiplication into addition.

92. Assertion: log₂(4 + 8) = log₂4 + log₂8.
Reason: The product law applies only to multiplication, not addition.

93. Assertion: A logarithm with base 1 is not valid in the real-number system used here.
Reason: Powers of 1 cannot produce the range of positive values required for a logarithmic inverse.

M. Higher-Order Thinking Questions

94. Without calculating directly, determine which is larger:

log₂ 8 or log₂ 16.

Explain your reasoning.

95. If log₃ x = 4, determine x without using a calculator.

96. A student says:

“Since log₂ 8 = 3, therefore log₈ 2 = 1/3.”

Is the statement correct? Explain.

97. If log_b x = p and log_b y = q, express log_b(xy) in terms of p and q.

98. If log_b x = p, express log_b(x⁴) in terms of p.

99. If log_b x = 3 and log_b y = 2, find:

log_b(x/y).

100. Find a value of x satisfying:

log₂ x + log₂ x = 6.

Answer Key

MCQs

  1. B
  2. B
  3. A
  4. C
  5. B
  6. B
  7. B
  8. B
  9. C
  10. B
  11. B
  12. C

Fill in the Blanks

  1. exponentiation
  2. x
  3. 0
  4. 1
  5. base
  6. positive
  7. 1
  8. log_b N
  9. log_b N
  10. n
  11. common
  12. addition

True/False

  1. True
  2. False
  3. True
  4. False
  5. False
  6. True
  7. False
  8. True
  9. False
  10. True

Match

  1. c
  2. a
  3. d
  4. e
  5. f
  6. b

Numerical Answers

  1. 4
  2. 4
  3. 3
  4. 4
  5. 0
  6. 1
  7. 64
  8. 125
  9. log₂16 = 4
  10. log₃243 = 5
  11. 4³ = 64
  12. 10² = 100
  13. log₇49 = 2
  14. 5⁴ = 625
  15. x = 16
  16. x = 9
  17. x = 125
  18. x = 8
  19. x = 27
  20. x = 25
  21. x = 100
  22. 1000; logarithmic value = 3
  23. 3
  24. 3 units
  25. 10,000
  26. 81
  27. 1
  28. x = 8