यदि `(X_(0),Y_(0))` समीकरणों `(2x)^(log 2)=(3y)^(log 3),3^(log x)=2^(log y)` का हल है तो `X_(0)` का मान है

Options

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

Answer

C

Explanation

`(2x)^(log 2)=(3y)^(log 3)....1`
`3^(log x)=2^(log y)....2`

समीकरण 1 से
`(2x) ^ (10g2 )=(3y)^ (log3)`

दोनों पक्षों में log लेने पर

` log (2x) ^ (log 2) = log (3y) ^ log 3`

` log_a (MN) = log_a M + log_a N`

`log 2 * (log 2 + log x) ``= log3 (log 3 + log y)....3 `

समीकरण 2 से
`3^(log x)=2^(log y)`
`3 ^( log x)`` = 2 ^ (log y)`
` log x . log 3``=log y.log 2`

` logy= (log x .log 3)/ log 2`

समीकरण 3 से
`log 2 * (log 2 + log x) ``= log3 (log 3 + log y)`

`log 2 * (log 2 + log x) ``= log3 (log 3 +(log x .log 3)/ log 2)`

`(log 2 )^ 2 (log 2+log x)``=(log 3)^ 2 (log 2+log x)`

`(log 2 )^ 2 (log 2+log x)``-(log 3)^ 2 (log 2+log x)=0`

`(log 2+log x )``( (log 2 )^ 2 - (log3)^ 2 )=0`

`log 2 + logx = 0 `,
`(log 2 )^ 2 ne (log 3) ^ 2`

`log2=-logx`

`log2=log(1/x)`

`x=2=x_0`