If `log x, log y and log z` are in AP, then

Options

A. `y^(2) = xz`
B. `x^(2) = yz`
C. `z^(2) = xy`
D. `y = xz`

Answer

To solve the problem where \( \log x, \log y, \) and \( \log z \) are in Arithmetic Progression (AP), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding AP**:
If three numbers \( a, b, c \) are in AP, then the condition is:
\[ 2b = a + c \]
Here, we can assign \( a = \log x \), \( b = \log y \), and \( c = \log z \).

2. **Applying the AP Condition**:
Substituting the values into the AP condition, we get:
\[ 2 \log y = \log x + \log z \]

3. **Using Logarithmic Properties**:
We can use the property of logarithms that states \( \log a + \log b = \log(ab) \). Therefore, we can rewrite the equation as:
\[ 2 \log y = \log(xz) \]

4. **Rewriting the Left Side**:
The left side can be rewritten using the property \( n \log a = \log(a^n) \):
\[ \log(y^2) = \log(xz) \]

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

### Final Result:
Thus, we conclude that if \( \log x, \log y, \) and \( \log z \) are in AP, then:
\[ y^2 = xz \]