Find the domain `f(x)=sqrt(log_(10){(log_(10)x)/(2(3-log_(10)x))})`

Options

A. `(10^(2), 10^(3)) `
B. `[10^(2), 10^(3)]`
C. ` [ 10^(2), 10^(3))`
D. `(10, 10^(3))`

Answer

A

Explanation

We, have
`f(x) = sqrt(log_(10){(log_(10)x)/(2(3-log_(10)x))})`
Clearly, f(x) is defined, if
`rArr { (log_(10)x)/(2-log _(10)x)} gt 0 (log_(10)x)/(2(3-log_(10)x)) gt 0 ` and `x gt 0`
`rArr (log_(10)x)/(2(3-log_(10)x))gt 1, (log_(10)x)/(3-log_(10)x) gt 0 and x gt 0`
`rArr (3(log_(10)x-2))/(2(log_(10) x-3)) le 0 ,(log_(10))/(log_(10)x-3) lt 0 and x gt 0`
`2le log_(10) x lt3, 0 ltlog_(10) x lt 3 and x gt 0`
`rArr 10^(2) lex le10^(3), 10^(0) lt x lt 10^(3) and x gt 0`
`rArr 10^(2) le x lt 10^(3) rArr x in [ 10,10^(3))`