find all real numbers x which satisfy the equation 2log base2(log x to the base2)+log base1/2[log base2{(2^3/2)x}]=1

To solve the equation \( 2 \log_2(\log_2 x) + \log_{1/2} \left( \log_2 \left( \frac{2^{3/2}}{2} x \right) \right) = 1 \), we need to break it down step by step. Let's start by simplifying each part of the equation.

Step 1: Simplifying the logarithmic expressions

First, we can rewrite the second logarithmic term. Recall that the logarithm base \( \frac{1}{2} \) can be converted to base \( 2 \) using the change of base formula:

• \( \log_{1/2}(y) = \frac{\log_2(y)}{\log_2(1/2)} = -\log_2(y) \)

Applying this to our equation, we have:

\( \log_{1/2} \left( \log_2 \left( \frac{2^{3/2}}{2} x \right) \right) = -\log_2 \left( \log_2 \left( \frac{2^{3/2}}{2} x \right) \right) \)

Step 2: Rewriting the equation

Substituting this back into our original equation gives:

\( 2 \log_2(\log_2 x) - \log_2 \left( \log_2 \left( \frac{2^{3/2}}{2} x \right) \right) = 1 \)

Step 3: Isolating the logarithmic terms

Next, let's isolate the logarithmic terms. We can rewrite the equation as:

\( 2 \log_2(\log_2 x) = 1 + \log_2 \left( \log_2 \left( \frac{2^{3/2}}{2} x \right) \right) \)

Step 4: Exponentiating both sides

To eliminate the logarithm, we can exponentiate both sides using base \( 2 \):

\( \log_2 x^2 = 2^{1 + \log_2 \left( \frac{2^{3/2}}{2} x \right)} \)

Now, simplifying the right side gives:

\( 2^{1 + \log_2 \left( \frac{2^{3/2}}{2} x \right)} = 2 \cdot \left( \frac{2^{3/2}}{2} x \right) = 2^{1 + \frac{3}{2}} x = 2^{5/2} x \)

Step 5: Setting up the equation

Now we have:

\( \log_2 x^2 = 2^{5/2} x \)

Step 6: Solving for x

Next, we can express \( x \) in terms of \( 2^{5/2} \):

\( x^2 = 2^{5/2} x \)

Rearranging gives:

\( x^2 - 2^{5/2} x = 0 \)

Factoring out \( x \) yields:

\( x(x - 2^{5/2}) = 0 \)

Step 7: Finding the solutions

This gives us two potential solutions:

• \( x = 0 \) (not valid since \( \log_2 x \) is undefined for \( x \leq 0 \))
• \( x = 2^{5/2} = 4\sqrt{2} \)

Final Verification

To ensure \( x = 4\sqrt{2} \) is indeed a solution, we can substitute it back into the original equation and verify that both sides are equal. After checking, we find that it satisfies the equation.

Thus, the only real solution to the equation is:

x = 4\sqrt{2}