The number of real roots of the equation
`1+3^(x//2)=2^(x)`, is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of real roots of the equation \( 1 + 3^{\frac{x}{2}} = 2^x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 1 + 3^{\frac{x}{2}} = 2^x \]

### Step 2: Divide by \( 2^x \)
To simplify the equation, we can divide both sides by \( 2^x \):
\[ \frac{1}{2^x} + \frac{3^{\frac{x}{2}}}{2^x} = 1 \]
This can be rewritten as:
\[ \frac{1}{2^x} + \left(\frac{3^{\frac{1}{2}}}{2}\right)^x = 1 \]
Let \( a = \frac{3^{\frac{1}{2}}}{2} = \frac{\sqrt{3}}{2} \). Thus, we have:
\[ \frac{1}{2^x} + a^x = 1 \]

### Step 3: Analyze the function
Define the function:
\[ f(x) = \frac{1}{2^x} + a^x \]
We need to find the number of solutions to the equation \( f(x) = 1 \).

### Step 4: Determine the behavior of \( f(x) \)
- As \( x \to -\infty \):
\[ f(x) \to \infty \quad (\text{since both } \frac{1}{2^x} \text{ and } a^x \text{ approach infinity}) \]
- As \( x \to \infty \):
\[ f(x) \to 0 \quad (\text{since both } \frac{1}{2^x} \text{ and } a^x \text{ approach zero}) \]

### Step 5: Check specific values of \( x \)
1. **At \( x = 0 \)**:
\[ f(0) = \frac{1}{2^0} + a^0 = 1 + 1 = 2 \]
2. **At \( x = 1 \)**:
\[ f(1) = \frac{1}{2^1} + a^1 = \frac{1}{2} + \frac{\sqrt{3}}{2} \approx 1.366 \quad (\text{greater than } 1) \]
3. **At \( x = 2 \)**:
\[ f(2) = \frac{1}{2^2} + a^2 = \frac{1}{4} + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1 \quad (\text{equals } 1) \]
4. **At \( x = 3 \)**:
\[ f(3) = \frac{1}{2^3} + a^3 = \frac{1}{8} + \left(\frac{\sqrt{3}}{2}\right)^3 = \frac{1}{8} + \frac{3\sqrt{3}}{8} \approx 0.649 \quad (\text{less than } 1) \]

### Step 6: Conclusion
Since \( f(x) \) is continuous and decreases from \( \infty \) to \( 0 \), and we found:
- \( f(0) > 1 \)
- \( f(2) = 1 \)
- \( f(3) < 1 \)

This indicates that there is exactly one solution at \( x = 2 \).

Thus, the number of real roots of the equation \( 1 + 3^{\frac{x}{2}} = 2^x \) is:
\[ \boxed{1} \]