If `log_3 2, log_3 (2^x -5)` and `log_3 (2^x-7/2)` are in `A.P`, determine the value of `x`.

Options

A. 2
B. 3
C. 4
D. `2,3`

Answer

B

Explanation

`:. log_(3)2,log_(3)(2^(x)-5)` and `log_(3)(2^(x)-(7)/(2))" are in AP. " " " "……..(i)"`
For defined, `2^(x)-5gt0` and `2^(x)-(7)/(2)gt0`
`:. 2^(x)gt5 " " "…(ii)"`
From Eq.(i),`2,2^(x)-5,2^(x)-(7)/(2)` are in GP. `:." " (2^(x)-5)^(2)=2*(2^(x)-(7)/(2))`
`implies 2^(2x)-12*2^(x)+32=0`
`implies (2^(x)-8)(2^(x)-4)=0`
`:." " 2^(x)=8,4`
`implies 2^(x)=8=2^(3),2^((x) ne4 " " [" from Eq. (ii) "]`.