I'm going to suggest that the "correct" answer is 4 and that the reason why is that there are two mistakes in the problem. x=4 is the correct answer to the question: Find the value of '$x$' if, $\large \left(\frac{1}{2^{\log_x 4}} \right) + \left( \frac{1}{2^{\log_x 16}} \right) + \left(\frac{1}{2^{\log_x 64}} \right) + \cdots = 2 $
So, I'm suggesting that it should both have been a sum rather than a product (since in its current form, it has no solution as Jonas correctly pointed out) and that the increasing series should have been $4^i$ rather than $4^{2^i}$. Although it can be solved if we just change it to a sum and leave it as $4^{2^i}$, it's difficult to solve and won't come out evenly (using Wolfram to calculate, the correct answer for just changing it to a sum is greater than 63, but less than 64). But if we change it to be 4,16,64,256,..., then we get the nice tidy answer of 4. This is probably what they were looking for, but they completely botched the question.