Find the domain of the following functions : `f(x)=sqrt(log_10((log_10x)/(2(3-log_10x)))`

Answer

`[100,1000)`

Explanation

`f(x)=sqrt(log_(10){(log_(10)x)/(2(3-log_(10)x))})`
Clearly, `f(x)` is defined if
`log_(10){(log_(10)x)/(2(3-log_(10)x))}ge 0,(log_(10)x)/(2(3-log_(10)x)) gt 0, " and " x gt 0`
or `(log_(10)x)/(2(3-log_(10)x)) ge 1, (log_(10)x)/(log_(10)x-3) lt 0 " and " x gt 0`
or `(3(log_(10)x-2))/(2(log_(10)x-3))le 0,(log_(10)x)/(log_(10)x-3) lt 0 " and " x gt 0`
or `2 le log_(10)x lt 3, 0 lt log_(10)x lt 3, " and " x gt 0`
or `10^(2) le x lt 10^(3), 10^(0) lt x lt 10^(3), " and " x gt 0`
or `x in [10^(2),10^(3))`