Given:

$A=\left[\begin{array}{lll}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 2& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 3\end{array}\right]$

$\Rightarrow A=\left[\begin{array}{lll}1& \frac{\log y}{\log x}& \frac{\log z}{\log x}\\ \frac{\log x}{\log y}& 2& \frac{\log z}{\log y}\\ \frac{\log x}{\log z}& \frac{\log y}{\log z}& 3\end{array}\right]$

$\Rightarrow A=\frac{1}{\log x\log y\log z}\left[\begin{array}{lll}\log x& \log y& \log z\\ \log x& 2\log y& \log z\\ \log x& \log y& 3\log z\end{array}\right]$

$\Rightarrow \left|A\right|=\frac{1}{\log x·\log y·\log z}\left|\begin{array}{lll}\log x& \log y& \log z\\ \log x& 2\log y& \log z\\ \log x& \log y& 3\log z\end{array}\right|$

$\Rightarrow \left|A\right|=\frac{\log x·\log y·\log z}{\log x·\log y·\log z}\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 1& 1& 3\end{array}\right|$

$\Rightarrow \left|A\right|=\left|\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 0& 2\end{array}\right|$

$\Rightarrow \left|A\right|=2$

Now,

$\left|adj\left(adj{A}^2\right)\right|={\left|adj{A}^2\right|}^2={\left|A^2\right|}^4={\left|A\right|}^8=2^8$