$Thedomainoffunction{\log }_{10}{\log }_{10}{\log }_{10}{\log }_{10}{\log }_{10}^{x}is$

$\begin{array}{l}f\left(x\right)={\log }_{10}.{\log }_{10}.{\log }_{10}.{\log }_{10}.{\log }_{10}^{x}isdefinediff\\ {\log }_{10}.{\log }_{10}.{\log }_{10}.{\log }_{10}^x>0\\ \Rightarrow {\log }_{10}.{\log }_{10}.{\log }_{10}^x>{10}^0\\ \Rightarrow {\log }_{10}.{\log }_{10}.{\log }_{10}^x>1\\ \Rightarrow {\log }_{10}.{\log }_{10}^x>10\\ \Rightarrow {\log }_{10}^x={10}^{10}\\ \Rightarrow x>{10}^{{10}^{10}}\\ \therefore x\in \left({10}^{{10}^{10}},\infty \right)\end{array}$