Find all real numbers x which satisty the equation. `2log_2log_2x+log_(1/2)log_2(2sqrt2x)=1`

Answer

To solve the equation \( 2 \log_2 (\log_2 x) + \log_{\frac{1}{2}} (\log_2 (2\sqrt{2}x)) = 1 \), we will proceed step by step.

### Step 1: Rewrite the logarithm
First, we can rewrite the logarithm with base \( \frac{1}{2} \) in terms of base 2:
\[ \log_{\frac{1}{2}} y = -\log_2 y \]
Thus, the equation becomes:
\[ 2 \log_2 (\log_2 x) - \log_2 (\log_2 (2\sqrt{2}x)) = 1 \]

### Step 2: Simplify the logarithm
Next, simplify \( \log_2 (2\sqrt{2}x) \):
\[ \log_2 (2\sqrt{2}x) = \log_2 (2) + \log_2 (\sqrt{2}) + \log_2 (x) = 1 + \frac{1}{2} + \log_2 (x) = \frac{3}{2} + \log_2 (x) \]
Substituting this back into the equation gives:
\[ 2 \log_2 (\log_2 x) - \log_2 \left( \frac{3}{2} + \log_2 x \right) = 1 \]

### Step 3: Isolate the logarithm
Rearranging the equation:
\[ 2 \log_2 (\log_2 x) = 1 + \log_2 \left( \frac{3}{2} + \log_2 x \right) \]
Using properties of logarithms, we can express \( 2 \log_2 (\log_2 x) \) as:
\[ \log_2 ((\log_2 x)^2) \]
So we have:
\[ \log_2 ((\log_2 x)^2) = 1 + \log_2 \left( \frac{3}{2} + \log_2 x \right) \]

### Step 4: Combine logarithms
Using the property \( \log_a b + \log_a c = \log_a (bc) \):
\[ \log_2 ((\log_2 x)^2) = \log_2 (2 \cdot \left( \frac{3}{2} + \log_2 x \right)) \]
This implies:
\[ (\log_2 x)^2 = 2 \left( \frac{3}{2} + \log_2 x \right) \]

### Step 5: Rearranging the equation
Expanding and rearranging gives:
\[ (\log_2 x)^2 - 2 \log_2 x - 3 = 0 \]

### Step 6: Solve the quadratic equation
Let \( t = \log_2 x \). The equation becomes:
\[ t^2 - 2t - 3 = 0 \]
Factoring gives:
\[ (t - 3)(t + 1) = 0 \]
Thus, \( t = 3 \) or \( t = -1 \).

### Step 7: Find \( x \)
Now, converting back to \( x \):
1. If \( t = 3 \):
\[ \log_2 x = 3 \implies x = 2^3 = 8 \]
2. If \( t = -1 \):
\[ \log_2 x = -1 \implies x = 2^{-1} = \frac{1}{2} \]

### Step 8: Check for validity
Since \( \log_2 (\log_2 x) \) must be greater than 0, we check:
- For \( x = 8 \): \( \log_2 8 = 3 \) (valid)
- For \( x = \frac{1}{2} \): \( \log_2 \frac{1}{2} = -1 \) (not valid)

### Final Answer
Thus, the only valid solution is:
\[ \boxed{8} \]