If x,y,z are in G.P. then log x , log y, logz, are in

Options

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

Answer

To solve the problem, we need to show that if \( x, y, z \) are in a geometric progression (G.P.), then \( \log x, \log y, \log z \) are in an arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the G.P. Condition**:
Since \( x, y, z \) are in G.P., we have the relationship:
\[ y^2 = x \cdot z \]

2. **Applying Logarithms**:
We take the logarithm of both sides of the equation:
\[ \log(y^2) = \log(x \cdot z) \]

3. **Using Logarithmic Properties**:
Using the property of logarithms that states \( \log(a^b) = b \cdot \log(a) \) and \( \log(a \cdot b) = \log(a) + \log(b) \), we can rewrite the equation:
\[ 2 \log y = \log x + \log z \]

4. **Rearranging the Equation**:
Now, we can rearrange the equation to express \( \log y \):
\[ \log y = \frac{\log x + \log z}{2} \]

5. **Identifying the A.P. Condition**:
The equation \( \log y = \frac{\log x + \log z}{2} \) indicates that \( \log y \) is the average of \( \log x \) and \( \log z \). This is the definition of an arithmetic progression (A.P.). Therefore, we can conclude that:
\[ \log x, \log y, \log z \text{ are in A.P.} \]

### Final Conclusion:
Thus, if \( x, y, z \) are in G.P., then \( \log x, \log y, \log z \) are in A.P.

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