Question

Statement-1: The solution set of the equation `"log"_(x) 2 xx "log"_(2x) 2 = "log"_(4x) 2 "is" {2^(-sqrt(2)), 2^(sqrt(2))}.` Statement-2 : `"log"_(b)a = (1)/("log"_(a)b) " and log"_(a) xy = "log"_(a) x + "log"_(a)y`
A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

Correct Answer - A
We have,
`"log"_(x) 2 xx "log"_(2x) 2 = "log"_(4x)2`
`rArr (1)/("log"_(2)x) xx (1)/("log"_(2)2x) = (1)/("log"_(2)4x)`
`rArr (1)/("log"_(2)x(1+"log"_(2)x)) = (1)/((2+"log"_(2)x))`
`rArr ("log"_(2)x)^(2) = 2 rArr "log"_(2) x = +- sqrt(2) rArr x = 2^(sqrt(2)), 2^(-sqrt(2))`
Hence, the solution set is `{2^(-sqrt(2)), 2^(sqrt(2))}`