Logarithms — Assertion-Reason Type Questions

Assertion (A): If $\log_x \space \dfrac{1}{8} = -\dfrac{1}{3}$, then the value of x is 2.

Reason (R): If n^x = m, then log_n m = x.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

$\Rightarrow \log_x \space \dfrac{1}{8} = -\dfrac{1}{3} \\[1em] \Rightarrow x^{-\dfrac{1}{3}} = \dfrac{1}{8} \\[1em] \Rightarrow \Big(\dfrac{1}{x}\Big)^{\dfrac{1}{3}} = \Big(\dfrac{1}{8}\Big) \\[1em] \text{ Cubing on both sides, we get:} \\[1em] \Rightarrow \Big[\Big(\dfrac{1}{x}\Big)^{\dfrac{1}{3}}\Big]^3 = \Big(\dfrac{1}{8}\Big)^3 \\[1em] \Rightarrow \Big(\dfrac{1}{x}\Big)^{\dfrac{1}{3} \times 3} = \Big(\dfrac{1}{512}\Big) \\[1em] \Rightarrow \dfrac{1}{x} = \dfrac{1}{512} \\[1em] \Rightarrow x = 512.$

So, Assertion (A) is false.

If n^x = m, then log_n m = x.

This is the fundamental definition of a logarithm.

So, Reason (R) is true.

A is false, R is true.

Hence, option 2 is the correct option.

Assertion (A): log (2 + 3 + 4) = log 2 + log 3 + log 4

Reason (R): log _x x = 0

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

log (2 + 3 + 4) = log 2 + log 3 + log 4

Simplifying L.H.S,

⇒ log (2 + 3 + 4) = log (9)

Simplifying R.H.S,

⇒ log 2 + log 3 + log 4 = log (2 × 3 × 4) = log (24)

Since, L.H.S ≠ R.H.S

So, Assertion (A) is false.

⇒ log_x x = 1 and not zero.

So, Reason (R) is false.

Both A and R are false.

Hence, option 4 is the correct option.