If `x^y = y^x and y = 2x` then x is equal to :

Options

A. 2
B. `-2`
C. `1`
D. `-1`

Answer

To solve the equation \(x^y = y^x\) given that \(y = 2x\), we can follow these steps:

### Step 1: Substitute \(y\) in the equation
We start with the equation:
\[ x^y = y^x \]
Since we know \(y = 2x\), we can substitute \(y\) in the equation:
\[ x^{2x} = (2x)^x \]

### Step 2: Expand the right side
Now, we can expand the right side:
\[ (2x)^x = 2^x \cdot x^x \]
So, we rewrite the equation as:
\[ x^{2x} = 2^x \cdot x^x \]

### Step 3: Divide both sides by \(x^x\)
Next, we divide both sides of the equation by \(x^x\):
\[ \frac{x^{2x}}{x^x} = \frac{2^x \cdot x^x}{x^x} \]
This simplifies to:
\[ x^{2x - x} = 2^x \]
or
\[ x^x = 2^x \]

### Step 4: Compare the bases
Now we have:
\[ x^x = 2^x \]
Since the bases are the same, we can equate the exponents:
\[ x = 2 \]

### Conclusion
Thus, the value of \(x\) is:
\[ \boxed{2} \]

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