To solve the relation \( 2 \log y - \log x - \log (y - 1) = 0 \), we will follow these steps:
### Step 1: Rewrite the equation using logarithmic properties
We start with the equation:
\[
2 \log y - \log x - \log (y - 1) = 0
\]
Using the property of logarithms, \( a \log b = \log b^a \), we can rewrite \( 2 \log y \) as \( \log y^2 \). Thus, the equation becomes:
\[
\log y^2 - \log x - \log (y - 1) = 0
\]
### Step 2: Combine the logarithms
Using the property that \( \log a - \log b = \log \left( \frac{a}{b} \right) \), we can combine the logarithms:
\[
\log \left( \frac{y^2}{x(y - 1)} \right) = 0
\]
### Step 3: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides:
\[
\frac{y^2}{x(y - 1)} = 1
\]
### Step 4: Rearrange the equation
Now, we rearrange the equation:
\[
y^2 = x(y - 1)
\]
Expanding the right side gives:
\[
y^2 = xy - x
\]
### Step 5: Rearrange into standard quadratic form
Rearranging this into standard quadratic form:
\[
y^2 - xy + x = 0
\]
### Step 6: Use the quadratic formula
To find \( y \), we apply the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = -x \), and \( c = x \):
\[
y = \frac{x \pm \sqrt{x^2 - 4 \cdot 1 \cdot x}}{2 \cdot 1}
\]
This simplifies to:
\[
y = \frac{x \pm \sqrt{x^2 - 4x}}{2}
\]
### Step 7: Determine the conditions for real \( y \)
For \( y \) to be real, the discriminant must be non-negative:
\[
x^2 - 4x \geq 0
\]
Factoring gives:
\[
x(x - 4) \geq 0
\]
This inequality holds when \( x \leq 0 \) or \( x \geq 4 \). However, since \( x \) must be positive for \( \log x \) to be defined, we conclude:
\[
x \geq 4
\]
### Step 8: Determine the range of \( y \)
We also have \( y \) in terms of \( x \):
\[
y = \frac{x \pm \sqrt{x(x - 4)}}{2}
\]
The expression under the square root must be non-negative, which we already established.
### Step 9: Find the valid range for \( y \)
Since \( y \) must also satisfy \( y > 1 \) for \( \log(y - 1) \) to be defined, we analyze the roots:
- The minimum value of \( y \) occurs when \( x = 4 \):
\[
y = \frac{4 \pm \sqrt{4(4 - 4)}}{2} = \frac{4}{2} = 2
\]
Thus, the range of \( y \) is \( y \geq 2 \).
### Conclusion
The domain of \( x \) is \( [4, \infty) \) and the range of \( y \) is \( [2, \infty) \).
