If a, b, c are in G.P, then `log_a x, log_b x, log_c x` are in

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding G.P.**:
Since \( a, b, c \) are in G.P., we have the relationship:
\[ \frac{b}{a} = \frac{c}{b} \]
This implies:
\[ b^2 = ac \]

2. **Define the logarithms**:
Let:
\[ P = \log_a x, \quad Q = \log_b x, \quad R = \log_c x \]

3. **Express \( a, b, c \) in terms of \( x \)**:
Using the definition of logarithms, we can express \( a, b, c \) as:
\[ a = x^{1/P}, \quad b = x^{1/Q}, \quad c = x^{1/R} \]

4. **Substituting into the G.P. relationship**:
Substitute \( a, b, c \) into the equation \( b^2 = ac \):
\[ (x^{1/Q})^2 = x^{1/P} \cdot x^{1/R} \]
This simplifies to:
\[ x^{2/Q} = x^{1/P + 1/R} \]

5. **Equating the exponents**:
Since the bases are the same (both are \( x \)), we can equate the exponents:
\[ \frac{2}{Q} = \frac{1}{P} + \frac{1}{R} \]

6. **Rearranging the equation**:
Rearranging gives us:
\[ \frac{1}{P}, \frac{1}{Q}, \frac{1}{R} \text{ are in arithmetic progression (A.P.)} \]

7. **Conclusion**:
Since the reciprocals \( \frac{1}{P}, \frac{1}{Q}, \frac{1}{R} \) are in A.P., it follows that \( P, Q, R \) are in harmonic progression (H.P.).

### Final Answer:
Thus, if \( a, b, c \) are in G.P., then \( \log_a x, \log_b x, \log_c x \) are in H.P.

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