1.

If `log_(10)2, log_(10)(2^(x)-1) and log_(10)(2^(x)+3)` are three consecutive terms of an A.P, then the value of x isA. 1B. `log_(5)2`C. `log_(2)5`D. `log_(10)5`

Answer» Correct Answer - C
`log_(10)2, log_(10)(2^(x)-1) and log_(10)(2^(x)+3)` are in A.P.
Hence, common difference will be same. `therefore log_(10)(2^(x)-1)-log_(10)2=log(2^(x)+3)-log_(10)(2^(x)-1)`
`therefore log_(10)((2^(x)-1)/(2))=log_(10)((2^(x)+3)/(2^(x)-1))`
`implies(2^(x)-1)/(2)=(2^(x)+3)/(2^(x)-1)`
`(2^(x)-1)^(2)=2(2^(x)+3)`
`2^(2x)-2^(x+1)+1=2^(x+1)+6`
`2^(2x)-2^(x+2)=5`
Let `2^(x)=y`, then
`y^(2)-4y-5=0`
`y^(2)-5y+y-5=0`
`y(y-5)+1(y-5)=0`
`y=-1, y=5`
`"Therefore, "2^(x)=5`
`x=log_(2)5`.


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