Let `(x_0, y_0)` be the solution of the following equations In `(2x)^(In2)=(3y)^(ln3)`and `3^(lnx)=2^(lny)` Then `x_0` is

Options

A. `1/6`
B. ` 1/3`
C. `1/2`
D. 6

Answer

C

Explanation

` (2x)^("In "2) = (3y)^("In "3)` ...(i)
` 3^(" In "x) = 2^("In "y)` ...(ii)
` rArr (log x) ( log 3 ) = (log y ) log 2 `
` rArr log y = ((log x)(log 3))/(log 2) ` ...(iii)
In (i) taking log on both sides, we get
` (log 2){log 2+ log x} = log 3 { log 3 + log y}`
` ( log 2)^(2) + (log 2 ) ( log x ) = (log 3 )^(2 ) + ((log3 ) ^( 2) ( log x ) )/(log 2 ) ` [from (iii)]
` or ( log 2 ) ^(2) - ( log 3)^(2) = ((log 3)^(2) - (log 2)^(2))/(log 2 ) (log x ) `
` or - log 2 = log x `
` rArr x = 1/2 `
` rArr x _(0) = 1/2 *`