Find the domain of definition of function f(x) given by

Question

f(x)= log base4{log base5(log base3(18x - x^2 - 77)}

Answer 1

Let u=18x-x^2-77

log_3(u) exists if u>0

log_5(log_3)u)) exists if log_3(u)>0, or u > 1

log_4(log_5(log_3(u))) exists if log_5(log_3(u)) > 0, or log_3(u) > 1, or u > 3

So, we need
18x-x^2-77 > 3
18x-x^2-80 > 0
(10-x)(x-8) > 0
8 < x < 10