Solve for x and y, if `x gt 0 and y gt 0` :
`log xy = "log" (x)/(y) + 2 log 2 = 2`.

Answer

To solve for \( x \) and \( y \) given the equation:

\[ \log(xy) = \log\left(\frac{x}{y}\right) + 2 \log 2 = 2 \]

where \( x > 0 \) and \( y > 0 \), we can follow these steps:

### Step 1: Rewrite the equation using logarithmic properties

Using the property of logarithms that states \( \log(ab) = \log a + \log b \), we can rewrite the left side:

\[ \log(xy) = \log x + \log y \]

The right side can be simplified using the property \( \log\left(\frac{a}{b}\right) = \log a - \log b \):

\[ \log\left(\frac{x}{y}\right) = \log x - \log y \]

Thus, we can rewrite the equation as:

\[ \log x + \log y = \log x - \log y + 2 \log 2 \]

### Step 2: Simplify the equation

Now, we can simplify the equation further:

\[ \log x + \log y - \log x + \log y = 2 \log 2 \]

This simplifies to:

\[ 2 \log y = 2 \log 2 \]

### Step 3: Divide both sides by 2

Dividing both sides by 2 gives:

\[ \log y = \log 2 \]

### Step 4: Solve for \( y \)

Using the property of logarithms that states if \( \log a = \log b \), then \( a = b \), we find:

\[ y = 2 \]

### Step 5: Substitute \( y \) back into the original equation to find \( x \)

Now we substitute \( y = 2 \) back into the original equation:

\[ \log(xy) = 2 \]

This becomes:

\[ \log(x \cdot 2) = 2 \]

### Step 6: Rewrite the equation in exponential form

Using the definition of logarithms, we can rewrite this as:

\[ x \cdot 2 = 10^2 \]

### Step 7: Solve for \( x \)

This simplifies to:

\[ x \cdot 2 = 100 \]

Dividing both sides by 2 gives:

\[ x = \frac{100}{2} = 50 \]

### Final Solution

Thus, the values of \( x \) and \( y \) are:

\[ x = 50, \quad y = 2 \]