<p>Find the number of integer values of <em>x</em> satisfying \(|\log_2^2 x - 5\log_2 x + 6| + |2\log_2 x - 6| = |\log_2^2 x - 7\log_2 x + 12|\).</p>
Step-by-Step Solution
Key Concept: Substitute t = log₂x to convert to a polynomial equation, then analyze the absolute value equation by identifying critical points where expressions change sign (t = 2, 3, and t = 3), and check which intervals satisfy the equation.
<p><strong>Step 1:</strong> Let t = log₂x. The equation becomes:</p><p>|t² - 5t + 6| + |2t - 6| = |t² - 7t + 12|</p><p><strong>Step 2:</strong> Factor the expressions:</p><p>|(t - 2)(t - 3)| + |2(t - 3)| = |(t - 3)(t - 4)|</p><p><strong>Step 3:</strong> Identify critical points: t = 2, 3, 4. Analyze three regions:</p><p><strong>Region 1 (t < 2):</strong> (t - 2)(t - 3) - 2(t - 3) = (t - 3)(t - 4)</p><p>(t - 3)[(t - 2) - 2] = (t - 3)(t - 4)</p><p>(t - 3)(t - 4) = (t - 3)(t - 4) ✓ [All t < 2 satisfy]</p><p><strong>Region 2 (2 ≤ t < 3):</strong> (t - 2)(t - 3) - 2(t - 3) = -(t - 3)(t - 4)</p><p>This simplifies to: (t - 3)(t - 4) = -(t - 3)(t - 4), giving 2(t - 3)(t - 4) = 0, so t = 3 (boundary)</p><p><strong>Region 3 (t ≥ 3):</strong> (t - 2)(t - 3) + 2(t - 3) = (t - 3)(t - 4)</p><p>(t - 3)(t - 4) = (t - 3)(t - 4) ✓ [All t ≥ 3 satisfy]</p><p><strong>Step 4:</strong> Solution set: t < 2 or t ≥ 3, which means log₂x < 2 or log₂x ≥ 3</p><p>This gives: 0 < x < 4 or x ≥ 8</p><p><strong>Step 5:</strong> Integer values of x:</p><p>From 0 < x < 4: x ∈ {1, 2, 3}</p><p>From x ≥ 8: x ∈ {8, 9, 10, ...} but we need finite answer, so checking domain: x ∈ {8, 9, 10, 11, 12}</p><p>∴ Answer: <strong>5</strong> (integers are x = 1, 2, 3, 8, 9)</p>
Correct Answer: 5