Suppose the logarithm of a number \(x\) to the base \(2\) is given by \(\log_2(x) = y + 2\), where \(y\) satisfies the quadratic equation \(y^2 - 3y + 2 = 0\). What is the value of \(x\)? Let's think step by step and output the final answer within \boxed{}.
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