Basic Mathematics & Logarithm
Properties of Logarithms
Grade 11
Question:
<p>If <em>x</em>, <em>y</em>, <em>z</em> are positive reals with <em>xyz</em> = 10<sup>81</sup> and
\[(\log_{10} x)(\log_{10} yz) + (\log_{10} y)(\log_{10} z) = 468,\]
find the value of
\[\sqrt{(\log_{10} x)^2 + (\log_{10} y)^2 + (\log_{10} z)^2}.\]</p>
Step-by-Step Solution
Key Concept: Recognize that the constraint xyz = 10^81 gives log₁₀(x) + log₁₀(y) + log₁₀(z) = 81. Use substitution a = log₁₀(x), b = log₁₀(y), c = log₁₀(z) to convert the given equation into algebraic form, then apply the identity (a+b+c)² = a²+b²+c² + 2(ab+bc+ca).
<p><strong>Step 1:</strong> Convert the exponential constraint. Since xyz = 10⁸¹, taking log₁₀ of both sides:</p><p>log₁₀(x) + log₁₀(y) + log₁₀(z) = 81</p><p><strong>Step 2:</strong> Substitute a = log₁₀(x), b = log₁₀(y), c = log₁₀(z). Then a + b + c = 81.</p><p><strong>Step 3:</strong> Rewrite the given equation. Note that yz = 10^(b+c), so:</p><p>a(b + c) + bc = 468</p><p><strong>Step 4:</strong> Expand using b + c = 81 - a:</p><p>a(81 - a) + bc = 468</p><p>81a - a² + bc = 468</p><p><strong>Step 5:</strong> Use the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca):</p><p>81² = a² + b² + c² + 2(ab + ac + bc)</p><p>6561 = a² + b² + c² + 2[a(b + c) + bc]</p><p>6561 = a² + b² + c² + 2(468)</p><p>6561 = a² + b² + c² + 936</p><p>a² + b² + c² = 5625</p><p><strong>Step 6:</strong> Calculate the final answer:</p><p>√(a² + b² + c²) = √5625 = 75</p><p>∴ Answer: <strong>75</strong></p>
Correct Answer: 75