If `"log"_(ax)x, "log"_(bx) x, "log"_(cx)x` are in H.P., where a, b, c, x belong to `(1, oo)`, then a, b, c are in

Options

A. A.P
B. G.P
C. H.P
D. none of these

Explanation

To solve the problem, we need to show that if \( \log_a x, \log_b x, \log_c x \) are in Harmonic Progression (H.P.), then \( a, b, c \) are in Geometric Progression (G.P.).

### Step-by-step Solution:

1. **Understanding H.P. Condition:**
Since \( \log_a x, \log_b x, \log_c x \) are in H.P., we can use the property of H.P. which states that for three numbers \( A, B, C \) to be in H.P., the following must hold:
\[ 2B = A + C \]
Here, let \( A = \log_a x \), \( B = \log_b x \), and \( C = \log_c x \). Thus, we have:
\[ 2 \log_b x = \log_a x + \log_c x \]

2. **Using the Change of Base Formula:**
We can use the change of base formula for logarithms, which states that:
\[ \log_a x = \frac{\log x}{\log a}, \quad \log_b x = \frac{\log x}{\log b}, \quad \log_c x = \frac{\log x}{\log c} \]
Substituting these into our equation gives:
\[ 2 \cdot \frac{\log x}{\log b} = \frac{\log x}{\log a} + \frac{\log x}{\log c} \]

3. **Simplifying the Equation:**
We can factor out \( \log x \) (assuming \( x > 1 \) so \( \log x \neq 0 \)):
\[ 2 \cdot \frac{1}{\log b} = \frac{1}{\log a} + \frac{1}{\log c} \]

4. **Finding a Common Denominator:**
To combine the right-hand side, we can find a common denominator:
\[ 2 \cdot \frac{1}{\log b} = \frac{\log b}{\log a \cdot \log b} + \frac{\log b}{\log c \cdot \log b} \]
This simplifies to:
\[ 2 \cdot \frac{1}{\log b} = \frac{\log b (\log c + \log a)}{\log a \cdot \log c} \]

5. **Cross-Multiplying:**
Cross-multiplying gives:
\[ 2 \log a \cdot \log c = \log b (\log a + \log c) \]

6. **Rearranging:**
Rearranging this equation leads us to:
\[ \log b^2 = \log a \cdot \log c \]
This indicates that:
\[ b^2 = ac \]

7. **Conclusion:**
The equation \( b^2 = ac \) is the condition for \( a, b, c \) to be in Geometric Progression (G.P.). Therefore, we conclude that \( a, b, c \) are in G.P.

### Final Answer:
Thus, \( a, b, c \) are in Geometric Progression (G.P.).

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