If x,y,z are in G.P and `a^x=b^y=c^z`,then

Options

A. `log_(b)a=log_(a)c`
B. `log_(c)b=log_(a)c`
C. `log_(b)a=log_(c)b`
D. none of these

Answer

To solve the problem step by step, we need to analyze the given conditions and use properties of logarithms and geometric progressions (G.P.).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given that \( x, y, z \) are in G.P. and that \( a^x = b^y = c^z \).

2. **Setting up the Equation**:
Since \( a^x = b^y = c^z \), we can denote this common value as \( k \). Thus, we have:
\[ a^x = k, \quad b^y = k, \quad c^z = k \]

3. **Taking Logarithms**:
Taking the logarithm of all three equations gives us:
\[ x \log a = \log k, \quad y \log b = \log k, \quad z \log c = \log k \]

4. **Expressing \( x, y, z \)**:
From the equations above, we can express \( x, y, z \) in terms of \( k \):
\[ x = \frac{\log k}{\log a}, \quad y = \frac{\log k}{\log b}, \quad z = \frac{\log k}{\log c} \]

5. **Using the G.P. Condition**:
Since \( x, y, z \) are in G.P., we know that:
\[ y^2 = xz \]

6. **Substituting the Values**:
Substituting the values of \( x, y, z \) into the G.P. condition:
\[ \left(\frac{\log k}{\log b}\right)^2 = \left(\frac{\log k}{\log a}\right) \left(\frac{\log k}{\log c}\right) \]

7. **Simplifying the Equation**:
This simplifies to:
\[ \frac{(\log k)^2}{(\log b)^2} = \frac{(\log k)^2}{\log a \cdot \log c} \]

8. **Cancelling \( (\log k)^2 \)**:
Assuming \( \log k \neq 0 \), we can cancel \( (\log k)^2 \) from both sides:
\[ \frac{1}{(\log b)^2} = \frac{1}{\log a \cdot \log c} \]

9. **Cross Multiplying**:
Cross multiplying gives us:
\[ \log a \cdot \log c = (\log b)^2 \]

10. **Rearranging the Expression**:
This can be rewritten as:
\[ \frac{\log a}{\log b} = \frac{\log b}{\log c} \]

11. **Final Result**:
This indicates that:
\[ \log_b a = \log_c b \]

### Conclusion:
Thus, the required relation is \( \log_b a = \log_c b \).