The number of integral solutions of `log_(9)(x+1).log_(2)(x+1)-log_(9)(x+1)-log_(2)(x+1)+1lt0` is

Options

A. 4
B. 5
C. 6
D. 7

Answer

C

Explanation

We have `log_(9)(x+1).log_(2)(x+1)-log_(9)(x+1)-log_(2)(x+1)+1lt0`
`rArr (log_(9)(x+1)-1)(log_(2)(x+1)-1)lt0`
`rArr (log_(9)((x+1)/(9)))(log_(2)((x+1)/(2)))lt0`
Case I : `log_(9)((x+1)/(9))gt 0`
`rArr (x+1)/(9)gt 1`
`rArr (x+1)/(9)gt 1`
`rArr x gt 8` ......(1)
`therefore log_(2)((x+1)/(2))lt 0`
`rArr 0 lt (x+1)/(2)lt 1`
`rArr -1x lt 1` ......(2)
`therefore` form (1) and (2), we get `x in phi`
Case II: `log_(9)((x+1)/(9))lt 0`
`rArr 0lt (x+1)/(9)lt 1`
`rArr -1lt x lt 8` .......(3)
`therefore log_(2)((x+1)/(2))gt 0`
`rArr (x+1)/(2)gt 1`
`therefore` from (3) and (4), `x in (1,8)`
So integeral values are `x=2,3,4,5,6,7`