The number of solutions for real x, which satisfy the equation `2log_2 log_2x+log_(1//2) log_2(2sqrt2x)=1`:

Options

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

Answer

To solve the equation \(2 \log_2 (\log_2 x) + \log_{\frac{1}{2}} (\log_2 (2\sqrt{2}x)) = 1\), we will follow these steps:

### Step 1: Rewrite the logarithm with base \(\frac{1}{2}\)
The term \(\log_{\frac{1}{2}} (\log_2 (2\sqrt{2}x))\) can be rewritten using the change of base formula:
\[ \log_{\frac{1}{2}} y = -\log_2 y \]
Thus, we can rewrite the equation as:
\[ 2 \log_2 (\log_2 x) - \log_2 (\log_2 (2\sqrt{2}x)) = 1 \]

### Step 2: Simplify \(\log_2 (2\sqrt{2}x)\)
Now, we simplify \(\log_2 (2\sqrt{2}x)\):
\[ \log_2 (2\sqrt{2}x) = \log_2 (2) + \log_2 (\sqrt{2}) + \log_2 (x) = 1 + \frac{1}{2} + \log_2 (x) = \frac{3}{2} + \log_2 (x) \]

### Step 3: Substitute back into the equation
Substituting this back into our equation gives:
\[ 2 \log_2 (\log_2 x) - \log_2 \left(\frac{3}{2} + \log_2 x\right) = 1 \]

### Step 4: Isolate one of the logarithmic terms
Rearranging the equation, we have:
\[ 2 \log_2 (\log_2 x) = 1 + \log_2 \left(\frac{3}{2} + \log_2 x\right) \]

### Step 5: Exponentiate both sides
Exponentiating both sides gives:
\[ (\log_2 x)^2 = 2 \cdot \left(\frac{3}{2} + \log_2 x\right) \]
This simplifies to:
\[ (\log_2 x)^2 = 3 + 2 \log_2 x \]

### Step 6: Rearrange into standard quadratic form
Rearranging this into standard quadratic form:
\[ (\log_2 x)^2 - 2 \log_2 x - 3 = 0 \]

### Step 7: Factor the quadratic equation
Factoring the quadratic equation:
\[ (\log_2 x - 3)(\log_2 x + 1) = 0 \]

### Step 8: Solve for \(\log_2 x\)
Setting each factor to zero gives:
1. \(\log_2 x - 3 = 0 \Rightarrow \log_2 x = 3 \Rightarrow x = 2^3 = 8\)
2. \(\log_2 x + 1 = 0 \Rightarrow \log_2 x = -1 \Rightarrow x = 2^{-1} = \frac{1}{2}\)

### Step 9: Conclusion
Thus, the solutions for \(x\) are \(x = 8\) and \(x = \frac{1}{2}\). Therefore, the number of solutions for real \(x\) is **2**.

---