The value of x for which `y=log_(2){-log_(1//2)(1+(1)/(x^(1//4)))-1}` is a real number are

Options

A. [0,1]
B. (0,1)
C. `[1,oo)`
D. none of these

Answer

To solve the equation \( y = \log_{2} \left( -\log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{4}}} \right) - 1 \right) \) and find the values of \( x \) for which \( y \) is a real number, we need to follow these steps:

### Step 1: Determine the conditions for the logarithm
The logarithm \( \log_{2}(A) \) is defined when \( A > 0 \). Therefore, we need:
\[ -\log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{4}}} \right) - 1 > 0 \]
This simplifies to:
\[ -\log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{4}}} \right) > 1 \]

### Step 2: Rearranging the inequality
Multiplying both sides by -1 (which reverses the inequality):
\[ \log_{\frac{1}{2}} \left( 1 + \frac{1}{x^{\frac{1}{4}}} \right) < -1 \]

### Step 3: Understanding the logarithm base
The logarithm \( \log_{\frac{1}{2}}(B) \) is negative when \( B < 1 \) because the base \( \frac{1}{2} < 1 \). Therefore, we need:
\[ 1 + \frac{1}{x^{\frac{1}{4}}} < \frac{1}{2^{-1}} = 2 \]

### Step 4: Simplifying the inequality
Subtracting 1 from both sides gives:
\[ \frac{1}{x^{\frac{1}{4}}} < 1 \]
This can be rewritten as:
\[ \frac{1}{x^{\frac{1}{4}}} - 1 < 0 \]

### Step 5: Further manipulation
Rearranging gives:
\[ \frac{1 - x^{\frac{1}{4}}}{x^{\frac{1}{4}}} < 0 \]
This inequality holds when the numerator and denominator have opposite signs.

### Step 6: Analyzing the signs
1. **Denominator**: \( x^{\frac{1}{4}} > 0 \) when \( x > 0 \).
2. **Numerator**: \( 1 - x^{\frac{1}{4}} < 0 \) implies \( x^{\frac{1}{4}} > 1 \), or \( x > 1 \).

### Step 7: Combining the conditions
Thus, we have:
- \( x > 0 \) (from the denominator)
- \( x > 1 \) (from the numerator)

### Conclusion
The values of \( x \) for which \( y \) is a real number are:
\[ x > 1 \]