Find the domain of function `f(x)=(log)_4[(log)_5{(log)_3(18 x-x^2-77}]`

Explanation

f(x) is defined if
`log_(5){log_(3)(18-x^(2)-77)} gt 0 " and " 18x-x^(2)-77 gt 0`
or `log_(3)(18x-x^(2)-77) gt 5^(0) " and " x^(2)-18x+77 lt 0`
or `log_(3)(18x-x^(2)-77) gt 1 " and " (x-11)(x-7) lt 0`
or `18x-x^(2) -77 gt 3^(1) " and " 7 lt x lt 11`
or `18x -x^(2)-80 gt 0 " and " 7 lt x lt 11`
or `x^(2) -18x +80 lt 0 " and " 7 lt x lt 11`
or `(x-10)(x-8) lt 0 " and " 7 lt x lt 11`
or `8 lt x lt 10 " and " 7 lt x lt 11`
or `8 lt x lt 10`
or `x in (8,10)`
Hence, the domain of f(x) is `(8,10)`.