For the equation `1 + log_(x) ((4 - x)/(10)) = (log_(10)(log_(10)p) - 1)log_(x)10`, which one of the following(s) is (are) correct?

Options

A. if `p = 10^(3)`, then the equation has two real solutions
B. if `p = 10^(4)`, then the equation has exactly solutions
C. if `p in (10^(4), oo)`, then the equation has no real solutions
D. if `p in (1, 10^(4))`, then the equation has two distinct real solutions.

Answer

B, C

Explanation

`1 + log_(x)((4 - x)/(10)) = (log_(10)(log_(10)p)-1)log_(x)10`
`(log_(10)x+log_(10)((4 - x)/(10)))/(log_(10)x) = (log_(10)((log_(10)p)/(10)))/(log_(10)x)`
`x((4 - x)/(10)) = (log_(10)p)/(10)x ne 1`
`x^(2) - 4x + log_(10)p = 0`
`x = (4 +- sqrt(16 - 4 xx 1 xx log_(10)p))/(2)`
`x = 2 +- sqrt(4 - log_(10)p)`
`(1)` if `p = 10^(3)` then equation becomes
`x^(2) - 4x + 3 = 0 (x - 3)(x - 1) = 0` but `x ne 1`
`(2)` if `p = 10^(4)`, then `x = 2` is only real solution.
(3) if `p gt 10^(4) rArr log_(10)p gt 4`
(4) if `1 lt p lt 10^(4) rArr 0 lt log_(10)p lt 4` then equation has two real distinct solutions but `p ne 10^(3)`