If `x^y=y^x` and `y = 3x`, then x is equal to

Options

A. `sqrt2`
B. `sqrt3`
C. 3
D. 2

Answer

To solve the equation \( x^y = y^x \) given that \( y = 3x \), we can follow these steps:

### Step 1: Substitute the value of \( y \)
We start with the equation \( x^y = y^x \) and substitute \( y = 3x \):
\[ x^{3x} = (3x)^x \]

### Step 2: Expand the right-hand side
Now, we can expand the right-hand side:
\[ (3x)^x = 3^x \cdot x^x \]
Thus, our equation becomes:
\[ x^{3x} = 3^x \cdot x^x \]

### Step 3: Simplify the equation
We can simplify the left-hand side:
\[ x^{3x} = x^x \cdot x^{2x} \]
So, we rewrite our equation as:
\[ x^{2x} = 3^x \]

### Step 4: Take logarithm of both sides
Taking logarithm on both sides gives us:
\[ \log(x^{2x}) = \log(3^x) \]
Using the property of logarithms, we can simplify this to:
\[ 2x \log x = x \log 3 \]

### Step 5: Divide both sides by \( x \) (assuming \( x \neq 0 \))
We can divide both sides by \( x \):
\[ 2 \log x = \log 3 \]

### Step 6: Solve for \( \log x \)
Now we can isolate \( \log x \):
\[ \log x = \frac{\log 3}{2} \]

### Step 7: Exponentiate to find \( x \)
Exponentiating both sides gives us:
\[ x = 10^{\frac{\log 3}{2}} = 3^{\frac{1}{2}} = \sqrt{3} \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{\sqrt{3}} \]