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

We are given that the terms $$ a_1, a_2, a_3, \ldots $$ are in a Geometric Progression (G.P.).

Let the first term of the G.P. be $$ a $$ and the common ratio be $$ r $$.

The general term of a G.P. is given by the formula:

$$ a_m = a \cdot r^{m-1} $$

Step 1: Applying Logarithms

Taking the natural logarithm on both sides of the general term equation:

$$ \log a_m = \log(a \cdot r^{m-1}) $$

$$ \log a_m = \log a + \log(r^{m-1}) $$

$$ \log a_m = \log a + (m-1) \log r $$

This expression shows that the logarithms of terms in a G.P. form an Arithmetic Progression (A.P.).

The common difference of this progression is $$ \log r $$.

Step 2: Substituting into the Determinant

We substitute these logarithmic expressions into the given determinant:

$$ \Delta = \begin{vmatrix} \log a + (n-1)\log r & \log a + n\log r & \log a + (n+1)\log r \\ \log a + (n+2)\log r & \log a + (n+3)\log r & \log a + (n+4)\log r \\ \log a + (n+5)\log r & \log a + (n+6)\log r & \log a + (n+7)\log r \end{vmatrix} $$

Step 3: Column Operations

To simplify the matrix, we apply the following column operations:

$$ C_2 \to C_2 - C_1 $$

$$ C_3 \to C_3 - C_2 $$

Let us calculate the changes for the columns:

For the second column minus the first column:

$$ (\log a + n\log r) - (\log a + (n-1)\log r) = \log r $$

$$ (\log a + (n+3)\log r) - (\log a + (n+2)\log r) = \log r $$

$$ (\log a + (n+6)\log r) - (\log a + (n+5)\log r) = \log r $$

For the third column minus the second column:

$$ (\log a + (n+1)\log r) - (\log a + n\log r) = \log r $$

$$ (\log a + (n+4)\log r) - (\log a + (n+3)\log r) = \log r $$

$$ (\log a + (n+7)\log r) - (\log a + (n+6)\log r) = \log r $$

Substituting these simplified values back into the determinant gives:

$$ \Delta = \begin{vmatrix} \log a + (n-1)\log r & \log r & \log r \\ \log a + (n+2)\log r & \log r & \log r \\ \log a + (n+5)\log r & \log r & \log r \end{vmatrix} $$

Step 4: Evaluation

We can observe that the second column and the third column are completely identical.

According to the properties of determinants, if any two rows or columns of a determinant are identical, the total value of the determinant is zero.

Therefore, the final value is: $$ \Delta = 0 $$