The domain of the function
`f(x)=log_(e){log_(|sinx|)(x^(2)-8x+23)-(3)/(log_(2)|sinx|)}`
contains which of the following interval (s) ?

Options

A. `(3, pi)`
B. `(pi,(3pi)/(2))`
C. `((3pi)/(2),5)`
D. None of these

Answer

A, B, C

Explanation

`f(x)` is defined if `log_(|sinx|)(x^(2)-8x+23)-(3)/(log_(2)|sinx|) gt 0`
`or log_(|sinx|)((x^(2)-8x+23)/(8)) gt 0`
This is true if `|sinx| ne 0,1 and (x^(2)-8x+23)/(8) lt 1.`
Now, `(x^(2)-8x+23)/(8) lt 1 or x^(2)-8x+15 lt 0`
`or x in (3,5) -{pi,(3pi)/(2)}`
Domain `=(3,pi) cup (pi,(3pi)/(2)) cup ((3pi)/(2),5).`