The domain of the definition of the function $f\left(x\right)={\log }_4\left({\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)\right)\text{ is }$

$\begin{array}{l}\text{ Since }\log x\text{ is defined for }x>0\text{ . }\\ \text{ Therefore, for }f\left(x\right)={\log }_4\left({\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)\right.\end{array}$

$\begin{array}{l}{\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)>0\\ \Rightarrow {\log }_3\left(18x-x^2-77\right)>5^0\\ \Rightarrow \left(18x-x^2-77\right)>3^1\end{array}$

$\begin{array}{l}\Rightarrow x^2-18x+80<0\\ \Rightarrow (x-8)(x-10)<0\\ \Rightarrow 8<x<10\Rightarrow x\in (8,10)\end{array}$

Hence, the domain of definition of the given function is (8, 10).