If $a_1,a_2,a_3,....,a_n,....$ are in G.P., then the value of the determinant
$$\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$$
is

Let the G.P. be defined by the first term $A$ and common ratio $r.$ Then $a_k=A{r}^{k-1}.$
Taking logs, $\log a_k=\log A+(k-1)\log r$.
The determinant elements become terms of an Arithmetic Progression.
Apply column transformations $C_3\to C_3-C_2$ and $C_2\to C_2-C_1$.
The transformed determinant will have:
$\log a_{n+1}-\log a_n=\log (a_{n+1}/a_n)=\log r$.
$\log a_{n+2}-\log a_{n+1}=\log (a_{n+2}/a_{n+1})=\log r$.
The determinant becomes:
$$\left|\begin{array}{ccc}\log a_n& \log r& \log r\\ \log a_{n+3}& \log r& \log r\\ \log a_{n+6}& \log r& \log r\end{array}\right|$$
.
Since two columns are identical, the value of the determinant is $0$.