<p>If \(x = \log_2\!\left(\sqrt{56 + \sqrt{56 + \sqrt{56 + \cdots}}}\right)\), then which of the following statements is correct?</p>
Step-by-Step Solution
Key Concept: Evaluate the infinite radical first. Let y = sqrt(56 + y). Then y^2 - y - 56 = 0, so y = 8. Hence x = log_2 8 = 3, which lies between 2 and 4.
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Evaluate the infinite radical first. Let y = sqrt(56 + y). Then y^2 - y - 56 = 0, so y = 8. Hence x = log_2 8 = 3, which lies between 2 and 4. Trap: The value is exactly 3, so choose the interval that contains 3, not the one with strict lower bound 3. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: C