If x and y are positive real numbers such that ` 2log(2y - 3x) = log x + log y," then find the value of " x/y`.

Explanation

To solve the equation \( 2 \log(2y - 3x) = \log x + \log y \), we will follow these steps:

### Step-by-Step Solution:

1. **Apply Logarithmic Properties**:
We start with the equation:
\[ 2 \log(2y - 3x) = \log x + \log y \]
Using the property of logarithms that states \( a \log b = \log(b^a) \) and \( \log a + \log b = \log(ab) \), we can rewrite the equation as:
\[ \log((2y - 3x)^2) = \log(xy) \]

2. **Remove the Logarithm**:
Since the logarithmic function is one-to-one, we can eliminate the logarithm by exponentiating both sides:
\[ (2y - 3x)^2 = xy \]

3. **Expand the Left Side**:
Expanding the left side gives:
\[ 4y^2 - 12xy + 9x^2 = xy \]

4. **Rearranging the Equation**:
Rearranging the equation to one side results in:
\[ 4y^2 - 12xy + 9x^2 - xy = 0 \]
This simplifies to:
\[ 4y^2 - 13xy + 9x^2 = 0 \]

5. **Substituting \( u = \frac{x}{y} \)**:
Let \( u = \frac{x}{y} \), which implies \( x = uy \). Substituting this into the equation gives:
\[ 4y^2 - 13(uy)y + 9(uy)^2 = 0 \]
Simplifying, we have:
\[ 4y^2 - 13uy^2 + 9u^2y^2 = 0 \]
Factoring out \( y^2 \) (since \( y \neq 0 \)):
\[ y^2(4 - 13u + 9u^2) = 0 \]
Thus, we need to solve:
\[ 9u^2 - 13u + 4 = 0 \]

6. **Using the Quadratic Formula**:
The quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) applies here, where \( a = 9, b = -13, c = 4 \):
\[ u = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 9 \cdot 4}}{2 \cdot 9} \]
Calculating the discriminant:
\[ u = \frac{13 \pm \sqrt{169 - 144}}{18} = \frac{13 \pm \sqrt{25}}{18} = \frac{13 \pm 5}{18} \]
This gives us two potential solutions:
\[ u = \frac{18}{18} = 1 \quad \text{and} \quad u = \frac{8}{18} = \frac{4}{9} \]

7. **Evaluating the Solutions**:
Since \( u = \frac{x}{y} \), we have two potential values: \( u = 1 \) and \( u = \frac{4}{9} \). However, if \( u = 1 \), then \( x = y \) which leads to \( 2y - 3y = -y \), and the logarithm of a negative number is undefined. Therefore, we discard \( u = 1 \).

8. **Final Result**:
The only valid solution is:
\[ \frac{x}{y} = \frac{4}{9} \]