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

Let $a_k=a{r}^{k-1}$.$\log a_k=\log a+(k-1)\log r$.The entries are of the form $A+(k-1)D$, which is an A.P.The determinant is:
$$\left|\begin{array}{ccc}A+(n-1)D& A+nD& A+(n+1)D\\ A+(n+2)D& A+(n+3)D& A+(n+4)D\\ A+(n+5)D& A+(n+6)D& A+(n+7)D\end{array}\right|$$
Perform column operation $C_2\to C_2-C_1$ and $C_3\to C_3-C_2$:
$$\left|\begin{array}{ccc}A+(n-1)D& D& D\\ A+(n+2)D& D& D\\ A+(n+5)D& D& D\end{array}\right|$$
Since two columns are identical, the value of the determinant is $0$.