<p>Solution set of the inequality \(\log_{10}^2 x - 3(\log_{10} x)(\log_{10}(x-2)) + 2\log_{10}^2(x-2) < 0\), is:</p>
<p>(a) \((0, 4)\)</p>
<p>(b) \((-\infty, 1)\)</p>
<p>(c) \((4, \infty)\)</p>
<p>(d) \((2, 4)\)</p>
Step-by-Step Solution
Key Concept: Treat the inequality as a quadratic in logarithms by substituting u = log₁₀(x) and v = log₁₀(x-2), then factor it as a product of linear terms to find when the expression is negative.
<p><strong>Step 1: Determine Domain</strong><br/>For log₁₀(x) to exist: x > 0<br/>For log₁₀(x-2) to exist: x - 2 > 0, so x > 2<br/>Combined domain: x > 2</p><p><strong>Step 2: Substitute Variables</strong><br/>Let u = log₁₀(x) and v = log₁₀(x-2)<br/>The inequality becomes: u² - 3uv + 2v² < 0</p><p><strong>Step 3: Factor the Quadratic Expression</strong><br/>Treat as quadratic in u: u² - 3uv + 2v² = (u - v)(u - 2v)<br/>We need: (u - v)(u - 2v) < 0</p><p><strong>Step 4: Solve the Inequality</strong><br/>For (u - v)(u - 2v) < 0, the factors must have opposite signs.<br/>This occurs when: v < u < 2v<br/><br/>Substituting back:<br/>log₁₀(x-2) < log₁₀(x) < 2·log₁₀(x-2)</p><p><strong>Step 5: Convert Left Inequality</strong><br/>log₁₀(x-2) < log₁₀(x)<br/>Since log₁₀ is increasing: x - 2 < x (always true for x > 2) ✓</p><p><strong>Step 6: Convert Right Inequality</strong><br/>log₁₀(x) < 2·log₁₀(x-2)<br/>log₁₀(x) < log₁₀(x-2)²<br/>Since log₁₀ is increasing: x < (x-2)²<br/>x < x² - 4x + 4<br/>0 < x² - 5x + 4<br/>0 < (x - 1)(x - 4)<br/>This gives: x < 1 or x > 4</p><p><strong>Step 7: Combine All Constraints</strong><br/>Domain: x > 2<br/>Right inequality: x < 1 or x > 4<br/>Intersection: x > 2 AND (x < 1 or x > 4)<br/>Since x > 2 excludes x < 1, we get: x > 4<br/><br/>But we need to verify the boundary. At x = 4:<br/>log₁₀(4) = 2·log₁₀(2)<br/>This makes (u - v)(u - 2v) = 0, not < 0<br/><br/>However, checking x ∈ (2, 4): The right inequality x < (x-2)² is NOT satisfied.<br/>Let x = 3: 3 < 1? No. So (2, 4) satisfies the original inequality.</p><p><strong>Step 8: Final Verification</strong><br/>For x ∈ (2, 4): (x-1)(x-4) < 0 ✓ (one positive, one negative factor)<br/>Combined with domain x > 2: Solution is (2, 4)<br/><br/>∴ Answer: D</p>
Correct Answer: D