Let `(x_(0), y_(0))` be the solution of the following equations:
`(2x)^("In "2)=(3y)^("In "3)`
` 3^("In "x) = 2^("In "y)`
Then ` x_(0)` is

Options

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

Answer

C

Explanation

We have,
`(2x)^("In"2)=(3y)^("In"3)"and",3^("In"x)=2^("In"y) `
`rArrlog(2x)^("In"2)=log(3y)^("In"3)" and "log(3^("In"x))=log(2^("In"y))`
`rArr log2(log2+logx)=log3(log+logy)`
and, `log3 logx=log2 logy`
`rArr (log2)^(2)+(log2)(logx)=(log3)^(2)+((log3)^(2)logx)/(log2)`
[On eliminating logy]
`rArr ((log2)^(2)-(log3)^(2))/(log2)logx=(log3)^(2)-(log2)^(2)`
`rArr logx=-log2 rArrx=(1)/(2)`