the number of roots of the equation `log_(3sqrtx) x + log_(3x) (sqrtx) =0` is

Options

A. 1
B. 2
C. 3
D. 0

Answer

B

Explanation

`(log x)/(log 3 + (1//2)log x) + ((1//2)log x )/(log 3+ log x) = 0`
` rArr (log _(3) x)/(1+(1//2) log_(3) x) + 1/2 (log_(3)x)/((1+log_(3)x)) = 0`
Let `log_(3)x = y," we get "y/(1+(y//2)) + y/(2(1+y)) = 0`
` or y(2/(2+y)+1/(2(1+y)))=0`
` or y [4+4y+2+y]=0`
`rArr y = 0 or y =- 6//5`
` rArr log_(3) x = 0 or log_(3) x =- 6//5`
` rArr x = 1 or x = 3^(-6//5)`