If `(log)_2(5xx2^(1-x)+1)` , `log_4(2^(1-x)+1) ` and 1 are in A.P., then `x` equals a. `log_2 5` b. `1-log_5 2` c. `log_5 2` d. none of these

Options

A. `log_(2)5`
B. `1-log_(2)5`
C. `log_(5)2`
D. none of these

Answer

B

Explanation

The given number are in A.P.
`:." "2log_(4)(2^(1-x)+1)=log_(2)(5.2^(x)+1)+1`
`rArr" "2log_(2)2((2)/(2^(x))+1)=log_(2)(5.2^(x)+1)+log_(2)2`
`rArr" "(2)/(2)log_(2)((2)/(2^(x))+1)=log_(2)(5.2^(x)+1)2`
`rArr" "log_(2)((2)/(2^(x))+1)=log_(2)(10.2^(x)+2)`
`rArr" "(2)/(2^(x))+1=10.2^(x)+2`
`rArr" "(2)/(y)+1=10y+2," where"2^(x)=y`
`:." "10y^(2)+y-2=0`
`rArr" "(5y-2)(2y+1)=0`
`rArr" "y=2//5" "[becausey=2^(x)gt0]`
`rArr" "2^(x)=2//5rArrx=log_(2)(2//5)=log_(2)2-log_(2)5=1-log_(2)5`