KnowledgeBoat Logo
|

Mathematics

Assertion (A): log225log_{\sqrt{2}} 2^5 = 10.

Reason (R): log am bn = nm\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).

Logarithms

1 Like

Answer

Given,

log am bn

By changing the base, we get :

⇒ log bn log am⇒n log bm log a⇒nm log ab\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,

⇒log 225⇒log 25log 2⇒log 25log 212⇒5× log 212× log 2⇒101×1⇒10.\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.

Answered By

3 Likes

Related Questions