About Logarithms — Class 9 ICSE

Understand logarithmic notation, laws of logarithms, and solve logarithmic equations. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 7). On this page you can practice 57 questions across three difficulty levels — 18 easy, 19 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 34-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Logarithms

• Introduction to Logarithms: Unveiling the Exponent's Secret
• Fundamental Laws of Logarithms: Simplifying Expressions
• The Power Rule and Special Values: Masterful Simplification
• Solving Logarithmic Equations: Finding the Unknown Exponent
• Summary and Practice: Mastering Logarithms for ICSE

Interactive lesson · about 15 minutes · checkpoint question after every unit

Logarithms — solved examples for Class 9 ICSE

Practice questions on Logarithms

1. Q1.easy

   Consider the statement: 'For any positive base 'a' (where a ≠ 1), log_a(1) = 0.' Which of the following best explains why this statement is true?

   1. A)It is true because 1 raised to any power is 1.
   2. B)It is true because logarithms always start from 0.
   3. C)It is true because any number raised to the power of 0 is 1.
   4. D)It is true because 1 is the smallest positive number.
   Show answer

   Answer: It is true because any number raised to the power of 0 is 1.

   Hint: Convert the logarithmic statement log_a(1) = 0 into its equivalent exponential form and recall a fundamental property of exponents.

2. Q2.easy

   Rohan was asked to evaluate log₇(7). He wrote: 'log₇(7) = 0 because any number divided by itself is 1, and log(1) is 0.' Identify the mistake in Rohan's reasoning.

   1. A)The mistake is assuming log(1) is 0; it should be 1.
   2. B)There is no mistake; Rohan's reasoning is correct.
   3. C)The mistake is equating log₇(7) to division; it's an exponent.
   4. D)The mistake is in the final answer; log₇(7) should be 7.
   Show answer

   Answer: The mistake is equating log₇(7) to division; it's an exponent.

   Hint: Think about the definition of a logarithm: log_a(N) = x means a^x = N. What exponent do you need to raise 7 to get 7?

3. Q3.easy

   Which of the following statements correctly applies the product rule of logarithms?

   1. A)log(A × B) = log(A) + log(B)
   2. B)log(A + B) = log(A) + log(B)
   3. C)log(A × B) = log(A) × log(B)
   4. D)log(A × B) = log(A) / log(B)
   Show answer

   Answer: log(A × B) = log(A) + log(B)

   Hint: The product rule for logarithms converts multiplication inside the logarithm into addition of separate logarithms.

4. Q4.medium

   Which of the following statements is true?

   1. A)log_10(1) = 10
   2. B)log_5(5) = 0
   3. C)log_2(0) = 1
   4. D)log_7(1) = 0
   Show answer

   Answer: log_7(1) = 0

   Hint: Remember the special values of logarithms: log_b(1) and log_b(b).

5. Q5.medium

   Simplify log(25) + log(4). (Assume base 10)

   1. A)log(29)
   2. B)log(100)
   3. C)2
   4. D)10
   Show answer

   Answer: 2

   Hint: Apply the product rule of logarithms: log_b(M) + log_b(N) = log_b(MN).

6. Q6.medium

   Evaluate log_3(54) - log_3(2).

   1. A)log_3(52)
   2. B)3
   3. C)log_3(27)
   4. D)2
   Show answer

   Answer: 3

   Hint: Use the quotient rule of logarithms: log_b(M) - log_b(N) = log_b(M/N).

7. Q7.hard

   If 3^(log_x 81) = 9, what is the value of x?

   1. A)81
   2. B)3
   3. C)9
   4. D)27
   Show answer

   Answer: 9

   Hint: Express both sides of the equation with the same base and then equate exponents. Remember the definition of logarithm: log_b(N) = P means b^P = N.

8. Q8.hard

   If log_k(x), log_k(y), log_k(z) are in Arithmetic Progression (A.P.), which of the following statements must be true about x, y, z?

   1. A)x, y, z are in Arithmetic Progression
   2. B)x, y, z are in Harmonic Progression
   3. C)x, y, z are in Geometric Progression
   4. D)log_x(k), log_y(k), log_z(k) are in Arithmetic Progression
   Show answer

   Answer: x, y, z are in Geometric Progression

   Hint: Recall the property of numbers in A.P. and apply the logarithm laws to simplify the relationship between log_k(x), log_k(y), log_k(z).

9. Q9.hard

   Rahul was asked to solve the equation log₂(x) + log₂(x-2) = 3. His steps were:
   Step 1: log₂(x(x-2)) = 3
   Step 2: x(x-2) = 2³
   Step 3: x² - 2x = 8
   Step 4: x² - 2x - 8 = 0
   Step 5: (x-4)(x+2) = 0
   Step 6: x = 4 or x = -2
   Step 7: The solution set is {4, -2}.
   Which step contains the first error in Rahul's reasoning?

   1. A)Step 1
   2. B)Step 4
   3. C)Step 6
   4. D)Step 7
   Show answer

   Answer: Step 7

   Hint: Remember the fundamental domain restrictions for logarithms. The argument of a logarithm must always be positive.

These are 9 of the 57 questions available for Logarithms. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.