Logarithms — Assertion-Reason Type Questions

Assertion (A): log₂ 16 = 4.

Reason (R): log_a (bc) = log_a b + log_a c

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

log_a (bc) = log_a b + log_a c

This is a fundamental property of logarithms, known as the product rule. It states that the logarithm of a product is the sum of the logarithms of the individual factors.

∴ Reason (R) is true.

Given, log₂ 16

⇒ log₂ (4 x 4)

⇒ log₂ 4 + log₂ 4

⇒ log₂ 2² + log₂ 2²

⇒ 2log₂ 2 + 2log₂ 2

⇒ 2 + 2

⇒ 4.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): log₃ $\Big(\dfrac{1}{9}\Big)$ = -2.

Reason (R): log_a $\Big(\dfrac{1}{b}\Big)$ = -log_a b.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given,

⇒ log_a $\Big(\dfrac{1}{b}\Big)$

⇒ log_a b^-1

⇒ (-1) x log_a b

⇒ -log_a b

∴ Reason (R) is true.

Given, log₃ $\Big(\dfrac{1}{9}\Big)$

⇒ log₃ $\Big(\dfrac{1}{3^2}\Big)$

⇒ log₃ 3^-2

⇒ -2log₃ 3

⇒ -2 x 1

⇒ -2.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): $log_{\sqrt{2}} 2^5$ = 10.

Reason (R): log _a^m bⁿ = $\dfrac{n}{m}$ log _a b.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given,

log _a^m bⁿ

By changing the base, we get :

$\Rightarrow \dfrac{\text{ log } b^n}{\text{ log } a^m}\\[1em] \Rightarrow \dfrac{n\text{ log } b}{m\text{ log } a}\\[1em] \Rightarrow \dfrac{n}{m} \text{ log } _a b$

∴ Reason (R) is true.

Given,

$\Rightarrow \text{log }_{\sqrt{2} } 2^5\\[1em] \Rightarrow \dfrac{\text{log } 2^5}{\text{log }\sqrt{2}} \\[1em] \Rightarrow \dfrac{\text{log } 2^5}{\text{log }2^{\dfrac{1}{2}}} \\[1em] \Rightarrow \dfrac{5 \times \text{ log } 2}{\dfrac{1}{2} \times \text{ log } 2} \\[1em] \Rightarrow \dfrac{10}{1} \times 1\\[1em] \Rightarrow 10.$

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): If log x = $\dfrac{\text{ log } 8}{\text{ log } 0.25}$, then x = -6.

Reason (R): If log _a b = log _a c, then b = c.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

According to reason,

If log _a b = log _a c, then b = c.

If the logarithms of two numbers are equal, and they share the same base, then the numbers themselves must be equal.

∴ Reason (R) is true.

According to assertion,

$\Rightarrow \text{log x} = \dfrac{\text{ log } 8}{\text{ log } 0.25}\\[1em] \Rightarrow \text{log x} = \dfrac{\text{ log } 8}{\text{ log } \Big(\dfrac{25}{100}\Big)}\\[1em] \Rightarrow \text{log x} = \dfrac{\text{ log } 2^3}{\text{ log } \Big(\dfrac{1}{4}\Big)}\\[1em] \Rightarrow \text{log x} = \dfrac{\text{ log } 2^3}{\text{ log } \Big(\dfrac{1}{2^2}\Big)}\\[1em] \Rightarrow \text{log x} = \dfrac{\text{ log } 2^3}{\text{ log } 2^{-2}}\\[1em] \Rightarrow \text{log x} = \dfrac{3\text{ log } 2}{-2\text{ log } 2}\\[1em] \Rightarrow \text{log x} = \dfrac{-3}{2} \\[1em] \Rightarrow x = (10)^{-\dfrac{3}{2}}.$

∴ Assertion (A) is false.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.