If `(sqrt(2))^(x)+(sqrt(3))^(x)=(sqrt(13))^(x//2)`, then the number of real values of x is

Options

A. 2
B. 4
C. 1
D. None of these

Explanation

To solve the equation \( (\sqrt{2})^x + (\sqrt{3})^x = (\sqrt{13})^{x/2} \), we can follow these steps:

### Step 1: Rewrite the equation
We can express the terms in the equation using exponents:
\[ (\sqrt{2})^x = 2^{x/2}, \quad (\sqrt{3})^x = 3^{x/2}, \quad (\sqrt{13})^{x/2} = 13^{x/4} \]
Thus, we rewrite the equation as:
\[ 2^{x/2} + 3^{x/2} = 13^{x/4} \]

### Step 2: Introduce a substitution
Let \( y = x/4 \). Then, we have \( x = 4y \). Substituting \( x \) in the equation gives:
\[ 2^{2y} + 3^{2y} = 13^y \]

### Step 3: Analyze the function
Define the function:
\[ f(y) = 2^{2y} + 3^{2y} - 13^y \]
We need to find the number of real roots of \( f(y) = 0 \).

### Step 4: Evaluate the function at specific points
1. **At \( y = 0 \)**:
\[ f(0) = 2^{0} + 3^{0} - 13^{0} = 1 + 1 - 1 = 1 \]
2. **At \( y = 1 \)**:
\[ f(1) = 2^{2} + 3^{2} - 13^{1} = 4 + 9 - 13 = 0 \]
3. **At \( y = 2 \)**:
\[ f(2) = 2^{4} + 3^{4} - 13^{2} = 16 + 81 - 169 = -72 \]

### Step 5: Analyze the behavior of the function
- \( f(0) = 1 \) (positive)
- \( f(1) = 0 \) (root)
- \( f(2) = -72 \) (negative)

Since \( f(y) \) is continuous, and it changes from positive at \( y = 0 \) to zero at \( y = 1 \), and then to negative at \( y = 2 \), we can conclude that there is at least one root in the interval \( (0, 2) \).

### Step 6: Check the derivative
To determine the number of roots, we can check the derivative \( f'(y) \):
\[ f'(y) = 2^{2y} \ln(2^2) + 3^{2y} \ln(3^2) - 13^y \ln(13) \]
Both \( 2^{2y} \) and \( 3^{2y} \) grow exponentially, while \( 13^y \) grows faster than both as \( y \) increases. Thus, \( f(y) \) will eventually become negative and remain so.

### Step 7: Conclusion
Since \( f(y) \) is continuous and we have established that it has one root in the interval \( (0, 2) \) and will not cross the x-axis again due to the behavior of the exponential functions, we conclude that there is exactly **one real solution** for \( y \).

### Final Answer
Thus, the number of real values of \( x \) is **1**.

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