Class 9 Logarithms Mock Test (ICSE)

Topics in Logarithms

• Meaning of logarithm
• Converting between exponential and logarithmic form
• Laws of logarithms
• Expanding and simplifying logarithmic expressions
• Solving logarithmic equations

Logarithms practice questions with answers

1. Q1Multiple choice·1 mark
   The value of $\log_{3} 81$ is:

   1. (a)$3$
   2. (b)$27$
   3. (c)$2$
   4. (d)$4$
   Show answer and step marks

   1. Correct option: (d) $4$+1

   Answer: (d)
2. Q2Multiple choice·1 mark
   Which value of $x$ satisfies $\log_{2}(x - 1) + \log_{2}(x + 1) = 3$, where $x > 1$?

   1. (a)$9$
   2. (b)$\sqrt{7}$
   3. (c)$-3$
   4. (d)$3$
   Show answer and step marks

   1. Correct option: (d) $3$+1

   Answer: (d)
3. Q3Assertion and reason·1 mark
   Assertion (A): $\log_{2} 32 = 5$. Reason (R): If $\log_{a} b = c$, then $a^{c} = b$.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer and step marks

   1. Correct option: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A), since $2^{5} = 32$ follows from the definition of the logarithm+1

   Answer: (a)
4. Q4Assertion and reason·1 mark
   Assertion (A): $\log_{5} 25 = 2$. Reason (R): Logarithms are useful for multiplying very large numbers.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer and step marks

   1. Correct option: (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).+1

   Answer: (b)
5. Q5Short answer·2 marks
   The statement $\log_{10} 12 - \log_{10} 4 = \log_{10} 3$ is true. (a) Name the law of logarithms that this statement illustrates. [1] (b) Write this law in general form, using $\log_a M$ and $\log_a N$. [1]
   Show answer and step marks

   1. The law illustrated is the quotient (division) law: the difference of two logarithms with the same base is the logarithm of the quotient+1
   2. $\log_a M - \log_a N = \log_a \dfrac{M}{N}$+1

   Answer: (a) Quotient (division) law of logarithms; (b) $\log_a M - \log_a N = \log_a \dfrac{M}{N}$
6. Q6Short answer·2 marks
   Without using a calculator, evaluate $\log_{2} 48 - \log_{2} 3$.
   Show answer and step marks

   1. Apply the quotient law $\log_a M - \log_a N = \log_a \dfrac{M}{N}$: $\log_{2} 48 - \log_{2} 3 = \log_{2} \dfrac{48}{3} = \log_{2} 16$+1
   2. Since $16 = 2^{4}$, $\log_{2} 16 = 4$+1

   Answer: 4

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