If 1,` log _(9) (3^(1 - x) + 2) and log_(3) (4.3^(x) -1) ` are A.P. then x is

Options

A. `log_(4)3`
B. `log_(3)4`
C. `1-log_(3)4`
D. `log_(3)0.25`

Answer

C

Explanation

Since 1, `log_(9)(3^(1-x)+2)andlog_(3)(4.3^(x)-1)` are in A.P
`:." "2log_(9)(3^(1-x)+2)=1+log_(3)(4.3^(x)-1)`
`rArr" "2log_(3)2(3^(1-x)+2)=log_(3)3+log_(3)(4.3^(x)-1)`
`rArr" "(2)/(2)log_(3)(3^(1-x)+2)=log_(3){(4.3^(x)-1)}`
`rArr" "(3^(1-x)+2)=3.(4.3^(x)-1)`
`rArr" "(3)/(y)+2=12y-3,"where "y=3^(x)`
`rArr" "12y^(2)-5y-3=0`
`rArr" "(3y+1)(4y-3)=0`
`rArr" "y=33//4" "[becausey=3^(x)gt0]`
`rArr" "3^(x)=(3)/(4)`
`rArr" "x=log_(3)(3//4)=log_(3)3-log_(3)4rArrx=1-log_(3)4`