If `x^(y)=y^(x)" and "x=2y`, then the values of x and y are (`x, y gt 0`)

Options

A. `x=4, y=2`
B. `x=3, y=2`
C. `x=1, y=1`
D. None of these

Answer

To solve the problem where \( x^y = y^x \) and \( x = 2y \), we will follow these steps:

### Step 1: Substitute \( x \) in terms of \( y \)
Given that \( x = 2y \), we can substitute \( x \) in the equation \( x^y = y^x \).

\[ (2y)^y = y^{2y} \]

### Step 2: Expand both sides
Now, we can expand both sides of the equation:

\[ 2^y \cdot y^y = y^{2y} \]

### Step 3: Rewrite the right side
The right side \( y^{2y} \) can be rewritten as:

\[ y^{2y} = y^y \cdot y^y \]

### Step 4: Set the equation
Now we have:

\[ 2^y \cdot y^y = y^y \cdot y^y \]

### Step 5: Divide both sides by \( y^y \)
Since \( y^y \) is positive (as \( y > 0 \)), we can divide both sides by \( y^y \):

\[ 2^y = y^y \]

### Step 6: Analyze the equation
We need to find values of \( y \) such that \( 2^y = y^y \).

### Step 7: Take logarithm on both sides
Taking the logarithm of both sides gives us:

\[ y \log(2) = y \log(y) \]

### Step 8: Divide by \( y \) (since \( y > 0 \))
We can divide both sides by \( y \):

\[ \log(2) = \log(y) \]

### Step 9: Solve for \( y \)
This implies:

\[ y = 2 \]

### Step 10: Substitute back to find \( x \)
Now, substituting \( y = 2 \) back into \( x = 2y \):

\[ x = 2 \cdot 2 = 4 \]

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

\[ (x, y) = (4, 2) \]

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