Suppose `a,b,c gt 0, x,y,z gt 1 and a,b,c` are in G.P. If `a^(logx)=b^(log y)=c^(logz)`, then

Options

A. x, y, z are in A.P.
B. x, y, z are in G.P.
C. x, y, z are in H.P.
D. none of these

Answer

To solve the problem step by step, we start with the given conditions:

1. **Given Conditions:**
- \( a, b, c > 0 \)
- \( x, y, z > 1 \)
- \( a, b, c \) are in Geometric Progression (G.P.)
- \( a^{\log x} = b^{\log y} = c^{\log z} \)

2. **Understanding G.P.:**
Since \( a, b, c \) are in G.P., we can express \( b \) in terms of \( a \) and \( c \):
\[ b = \sqrt{ac} \]

3. **Setting a Common Value:**
Let \( k = a^{\log x} = b^{\log y} = c^{\log z} \). This means:
\[ a^{\log x} = k \quad (1) \]
\[ b^{\log y} = k \quad (2) \]
\[ c^{\log z} = k \quad (3) \]

4. **Expressing \( b \) and \( c \):**
From equation (1):
\[ \log k = \log a \cdot \log x \quad \Rightarrow \quad k = 10^{\log a \cdot \log x} \]
From equation (2):
\[ \log k = \log b \cdot \log y \quad \Rightarrow \quad k = 10^{\log b \cdot \log y} \]
From equation (3):
\[ \log k = \log c \cdot \log z \quad \Rightarrow \quad k = 10^{\log c \cdot \log z} \]

5. **Equating the Expressions for \( k \):**
Since all three expressions for \( k \) are equal, we can write:
\[ \log a \cdot \log x = \log b \cdot \log y = \log c \cdot \log z \]

6. **Using the G.P. Property:**
Since \( b = \sqrt{ac} \), we can express \( \log b \):
\[ \log b = \frac{1}{2} (\log a + \log c) \]

7. **Substituting \( \log b \):**
Using the property of logarithms:
\[ \log a \cdot \log x = \frac{1}{2} (\log a + \log c) \cdot \log y \]
\[ \log b \cdot \log y = \frac{1}{2} (\log a + \log c) \cdot \log y \]
\[ \log c \cdot \log z = \log c \cdot \log z \]

8. **Analyzing Relationships:**
From the above equations, we can analyze the relationships between \( x, y, z \):
- If \( x, y, z \) were in Arithmetic Progression (A.P.), then \( 2y = x + z \).
- If \( x, y, z \) were in Harmonic Progression (H.P.), then \( \frac{2}{y} = \frac{1}{x} + \frac{1}{z} \).

9. **Conclusion:**
Since we derived that \( 2 \cdot 10^{\log b \cdot \log y} \neq 10^{\log a \cdot \log x} + 10^{\log c \cdot \log z} \), we conclude that:
- \( x, y, z \) are not in A.P.
- Consequently, \( x, y, z \) are not in H.P.
- Finally, \( x, y, z \) are also not in G.P.