<p>Positive number \(x\), \(y\) and \(z\) satisfy \(xyz = 10^{81}\) and \((\log_{10} x)(\log_{10} yz) + (\log_{10} y)(\log_{10} z) = 468\). Find \(\sqrt{(\log_{10} x)^2 + (\log_{10} y)^2 + (\log_{10} z)^2}\).</p>
Step-by-Step Solution
Key Concept: Substitute a = log₁₀x, b = log₁₀y, c = log₁₀z to convert the constraint xyz = 10⁸¹ into a + b + c = 81, then expand the given equation using logarithm properties to find a² + b² + c².
<p><strong>Step 1:</strong> Let a = log₁₀x, b = log₁₀y, c = log₁₀z (all real numbers).</p><p><strong>Step 2:</strong> From xyz = 10⁸¹, taking log₁₀: log₁₀x + log₁₀y + log₁₀z = 81, so <strong>a + b + c = 81</strong>.</p><p><strong>Step 3:</strong> Rewrite the given equation using logarithm properties:</p><p>(log₁₀x)(log₁₀yz) + (log₁₀y)(log₁₀z) = 468</p><p>a(b + c) + bc = 468</p><p>ab + ac + bc = 468</p><p><strong>Step 4:</strong> Use the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + ac + bc):</p><p>81² = a² + b² + c² + 2(468)</p><p>6561 = a² + b² + c² + 936</p><p>a² + b² + c² = 5625</p><p><strong>Step 5:</strong> Therefore:</p><p>√(a² + b² + c²) = √5625 = <strong>75</strong></p><p>∴ Answer: <strong>75</strong></p>
Correct Answer: 75