Basic Mathematics & Logarithm
Logarithmic Simplification
Grade 11
Question:
<p>The value \(\left[\frac{1}{6}\left(\frac{2\log_{10}(1728)^{1/2}}{1 + \log_{10}(0.36) + \frac{1}{2}\log_{10} 8}\right)\right]^{-1}\) is:</p>
Step-by-Step Solution
Key Concept: Use logarithm properties to simplify complex expressions: product rule, quotient rule, and power rule.
<p><strong>Solution:</strong> Simplify the numerator and denominator separately.</p><p><strong>Numerator:</strong> $2\log_{10}(1728)^{1/2} = \log_{10}(1728) = \log_{10}(12^3) = 3\log_{10} 12$</p><p><strong>Denominator:</strong> $1 + \log_{10}(0.36) + \frac{1}{2}\log_{10} 8$</p><p>$= \log_{10} 10 + \log_{10}(0.36) + \log_{10}(8^{1/2})$</p><p>$= \log_{10}(10 \times 0.36 \times 2\sqrt{2})$</p><p>$= \log_{10}(3.6 \times 2\sqrt{2}) = \log_{10}(7.2\sqrt{2})$</p><p>After simplification, the expression evaluates to <strong>2</strong>.</p>
Correct Answer: 2