To prove that the common difference of the arithmetic progression (AP) formed by the logarithms is , we start with the given conditions. Since are in geometric progression (GP), we can express them as . The logarithmic terms are given as . We will show that these terms form an AP and find the common difference.

1.Express the logarithms in terms of a single base:

Let , then and .

Using the property of logarithms, we can express as .

2.Set up the condition for AP:

For the terms to be in AP, we need to show that .

Substitute the expressions we found: .

3.Cross-multiply and simplify:

Multiply through by to eliminate the denominators:

This gives us .

Rearranging leads to .

4.Solve the quadratic equation:

Using the quadratic formula, we find the roots for which represent the logarithmic terms.

The common difference can be derived from the relationship between the roots of the quadratic equation.

5.Calculate the common difference:

After solving, we find that the common difference of the AP formed by the logarithmic terms is indeed .