If `log_(10)x, log_(10)y, log_(10)z` are in AP, then x,y,z are in:

Options

A. AP
B. GP
C. HP
D. None of these

Answer

To solve the problem, we need to show that if \( \log_{10} x, \log_{10} y, \log_{10} z \) are in Arithmetic Progression (AP), then \( x, y, z \) are in Geometric Progression (GP).

### Step-by-Step Solution:

1. **Understanding the Condition of AP**:
Since \( \log_{10} x, \log_{10} y, \log_{10} z \) are in AP, we can use the property of AP which states that for three numbers \( A, B, C \) in AP, the following holds:
\[ 2B = A + C \]
Here, let \( A = \log_{10} x \), \( B = \log_{10} y \), and \( C = \log_{10} z \).

2. **Applying the AP Condition**:
Applying the AP condition:
\[ 2 \log_{10} y = \log_{10} x + \log_{10} z \]

3. **Using Logarithmic Properties**:
We can use the property of logarithms that states \( \log_{10} a + \log_{10} b = \log_{10} (a \cdot b) \). Therefore, we can rewrite the equation as:
\[ 2 \log_{10} y = \log_{10} (x \cdot z) \]

4. **Rearranging the Equation**:
Now, we can express \( 2 \log_{10} y \) as:
\[ \log_{10} (y^2) = \log_{10} (x \cdot z) \]

5. **Equating the Arguments**:
Since the logarithms are equal, we can equate the arguments:
\[ y^2 = x \cdot z \]

6. **Conclusion**:
The equation \( y^2 = x \cdot z \) indicates that \( x, y, z \) are in Geometric Progression (GP). Thus, we conclude that if \( \log_{10} x, \log_{10} y, \log_{10} z \) are in AP, then \( x, y, z \) are in GP.

### Final Answer:
If \( \log_{10} x, \log_{10} y, \log_{10} z \) are in AP, then \( x, y, z \) are in GP.

---