If a.b and c are in GP, then `log_(ax) x. log_(bx) x. log_(cx) x` . Are in :

Options

A. (A)GP
B. (B)HP
C. (C)AP
D. (D)None of these

Answer

To solve the problem, we need to determine the relationship between the logarithmic expressions given that \(a\), \(b\), and \(c\) are in geometric progression (GP).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
Since \(a\), \(b\), and \(c\) are in GP, we can express this condition mathematically as:
\[ b^2 = ac \]

2. **Expressing Logarithmic Terms**:
We need to analyze the terms \( \log_{ax} x \), \( \log_{bx} x \), and \( \log_{cx} x \). We can rewrite these logarithmic expressions using the change of base formula:
\[ \log_{ax} x = \frac{\log x}{\log(ax)} = \frac{\log x}{\log a + \log x} \]
\[ \log_{bx} x = \frac{\log x}{\log(b) + \log(x)} \]
\[ \log_{cx} x = \frac{\log x}{\log(c) + \log(x)} \]

3. **Setting Up the Variables**:
Let:
\[ k_1 = \log_{ax} x, \quad k_2 = \log_{bx} x, \quad k_3 = \log_{cx} x \]
Thus, we have:
\[ k_1 = \frac{\log x}{\log a + \log x}, \quad k_2 = \frac{\log x}{\log b + \log x}, \quad k_3 = \frac{\log x}{\log c + \log x} \]

4. **Finding a Common Form**:
To simplify, we can express each \(k_i\) in terms of \( \log x \):
\[ k_1 = \frac{1}{\frac{\log a}{\log x} + 1}, \quad k_2 = \frac{1}{\frac{\log b}{\log x} + 1}, \quad k_3 = \frac{1}{\frac{\log c}{\log x} + 1} \]

5. **Using the GP Condition**:
Since \(b^2 = ac\), we can relate the logarithmic terms:
\[ \frac{1}{k_2} = \frac{1}{2} \left( \frac{1}{k_1} + \frac{1}{k_3} \right) \]
This implies that \(k_1\), \(k_2\), and \(k_3\) are in harmonic progression (HP).

6. **Conclusion**:
Therefore, we conclude that:
\[ \log_{ax} x, \log_{bx} x, \log_{cx} x \text{ are in HP.} \]

### Final Answer:
The logarithmic terms \( \log_{ax} x, \log_{bx} x, \log_{cx} x \) are in **Harmonic Progression (HP)**.