Find the domain of function f(x) = log_4[log_5log_3(18x-x^2-77)].

f(x) is defined if

$\log_5\{\log_3(18x-x^2-77)\} > 0$ and $18x-x^2-77>0$

or $\log_3 (18x-x^2-77) >5^0$ and $x^2-18x+77 <0$

or $\log_3 (18x-x^2-77) > 1$ and $(x - 11)(x-7)<0$

or $18x-x^2-77>3^1$ and $7 < x < 11$

or $18x-x^2-80> 0$ and $7 < x < 11$

or $x^2-18x + 80 < 0$ and $7 < x < 11$

or $(x-10)(x-8)<0$ and $7<x<11$

or $8<x<10$ and $7<x<11$

or $8 < x < 10$

or $x ∈ (8, 10)$

Hence, the domain off (x) is (8, 10).