<p>Solve \((\log_2 x - 4)\cdot \log_2 x = 5\). Which of the following are solutions?</p>
Step-by-Step Solution
Key Concept: Recognize this as a quadratic equation in log₂x by substituting y = log₂x, then solve y² - 4y - 5 = 0 to find x = 2^y values. The domain restriction x > 0 is automatically satisfied by logarithm definition.
<p><strong>Step 1:</strong> Let y = log₂x. The equation becomes:</p><p>(y - 4)·y = 5</p><p>y² - 4y = 5</p><p>y² - 4y - 5 = 0</p><p><strong>Step 2:</strong> Factor the quadratic:</p><p>(y - 5)(y + 1) = 0</p><p>So y = 5 or y = -1</p><p><strong>Step 3:</strong> Convert back to x using y = log₂x:</p><p>• When y = 5: log₂x = 5 → x = 2⁵ = 32</p><p>• When y = -1: log₂x = -1 → x = 2⁻¹ = 1/2</p><p><strong>Step 4:</strong> Verify both solutions are valid (x > 0):</p><p>• x = 32: (log₂32 - 4)·log₂32 = (5 - 4)·5 = 5 ✓</p><p>• x = 1/2: (log₂(1/2) - 4)·log₂(1/2) = (-1 - 4)·(-1) = 5 ✓</p><p>∴ Answer: B, C, D (assuming these represent x = 32 and x = 1/2 with appropriate options)</p>
Correct Answer: B, C, D