Solve `(x+1)^(log_(10) (x+1))=100(x+1)`

Options

A. all the roots are positive real numbers.
B. all the roots lie in the interval (0, 100)
C. all the roots lie in the interval [-1, 99]
D. none of these

Answer

C

Explanation

`(x+1)^(log_(10)(x+1))=100(x+1)`
` rArr log_(10) (x+1)^(log_(10)(x+1))=log_(10) (100(x+1))`
` rArr log_(10)(x+1)log_(10)(x+1) = 2 + log_(10) (x+1)`
Let ` log_(10)(x+1) = y`
` rArr y^(2)-y-2=0`
` rArr y = 2 or - 1`
` rArr log_(10)(x+1) = 2, -1`
` rArr x+1 = 100, 1//10`
` rArr x =99 or - 9//10`