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 equations \( x^y = y^x \) and \( x = 2y \), we will follow these steps:

### Step 1: Substitute \( x \) in terms of \( y \)
From the second equation, we have:
\[ x = 2y \]
Now, we can substitute \( x \) in the first equation \( x^y = y^x \):
\[ (2y)^y = y^{2y} \]

### Step 2: Simplify the equation
Now we can simplify the equation:
\[ (2^y)(y^y) = (y^2)^y \]
This can be rewritten as:
\[ (2^y)(y^y) = y^{2y} \]

### Step 3: Divide both sides by \( y^y \) (assuming \( y \neq 0 \))
We can divide both sides by \( y^y \):
\[ 2^y = y^{2y - y} = y^y \]

### Step 4: Rewrite the equation
Now we have:
\[ 2^y = y^y \]

### Step 5: Take the logarithm of both sides
Taking the logarithm of both sides gives:
\[ y \log(2) = y \log(y) \]

### Step 6: Divide both sides by \( y \) (assuming \( y \neq 0 \))
Assuming \( y \neq 0 \), we can divide both sides by \( y \):
\[ \log(2) = \log(y) \]

### Step 7: Solve for \( y \)
Exponentiating both sides gives:
\[ y = 2 \]

### Step 8: Find \( x \) using \( y \)
Now that we have \( y \), we can find \( x \) using the equation \( x = 2y \):
\[ x = 2(2) = 4 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ (x, y) = (4, 2) \]

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