If `log_(10)2,log_(10)(2^(x)-1)` and `log_(10)(2^(x)+3)` are in A.P then the value of x is

Options

A. more than two real x
B. no real x
C. exactly one real x
D. exactly two real x

Answer

C

Explanation

It is given that `log_(10)2,log_(10)(2^(x)-1)and log_(10)(2^(x)+3)` are three consecutive terms of an A.P.
`:." "2log_(10)(2^(x)-1)=log_(10)2+log_(10)(2^(x)+3)`
`rArr" "log_(10)(2^(x)-1)^(2)=log_(10)2(2^(x)+3)`
`rArr" "(2^(x)-1)^(2)=2(2^(x)+3)`
`rArr" "(2^(x))^(2)-4(2^(x))-5=0`
`rArr" "(2^(x)-5)(2^(x)+1)=0rArr2^(x)=5rArrx=log_(2)5`
Thus, given numbers are in A.P for exactly one real value of x.