Basic Mathematics & Logarithm
Properties of Logarithms
Grade 11
Question:
<p>Which of the following is/are correct?</p><p>(a) \(7^{-\log_7 6} + 81^{(1-\log_9 2)} = \frac{1}{6} + \frac{81}{4}\)</p><p>(b) \((1-\log_6 2)(1+\log_6 2)+(\log_6 2)^2 = 1\)</p><p>(c) \(\log_3 5 + \log_3 6 - \log_3 10 = 1\)</p><p>(d) \(\left(2^{\frac{1}{3}}+5^{\frac{1}{3}}\right)\left(2^{\frac{2}{3}}-2^{\frac{1}{3}}\cdot 5^{\frac{1}{3}}+5^{\frac{2}{3}}\right) = 7\)</p>
<p>(a) \(7^{-\log_7 6} + 81^{(1-\log_9 2)} = \frac{1}{6} + \frac{81}{4}\)</p>
<p>(b) \((1-\log_6 2)(1+\log_6 2)+(\log_6 2)^2 = 1\)</p>
<p>(c) \(\log_3 5 + \log_3 6 - \log_3 10 = 1\)</p>
<p>(d) \(\left(2^{\frac{1}{3}}+5^{\frac{1}{3}}\right)\left(2^{\frac{2}{3}}-2^{\frac{1}{3}}\cdot 5^{\frac{1}{3}}+5^{\frac{2}{3}}\right) = 7\)</p>
Step-by-Step Solution
Key Concept: Use logarithm properties: a^(-log_a x) = 1/x, and algebraic identities like (a+b)(a²-ab+b²) = a³+b³. Recognize that logarithmic expressions often simplify through product/quotient rules.
<p><strong>Option (a):</strong> 7^(-log₇6) + 81^(1-log₉2)</p><p>• 7^(-log₇6) = 1/6 ✓</p><p>• 81^(1-log₉2) = 81·81^(-log₉2) = 81·(9²)^(-log₉2) = 81·9^(-2log₉2) = 81·(2^(log₉9))^(-1) = 81/4 ✓</p><p>• Sum: 1/6 + 81/4 ✓ <strong>CORRECT</strong></p><p><strong>Option (b):</strong> (1-log₆2)(1+log₆2) + (log₆2)²</p><p>• (1-log₆2)(1+log₆2) = 1 - (log₆2)² (difference of squares)</p><p>• Add (log₆2)²: 1 - (log₆2)² + (log₆2)² = 1 ✓ <strong>CORRECT</strong></p><p><strong>Option (c):</strong> log₃5 + log₃6 - log₃10</p><p>• = log₃(5·6/10) = log₃(30/10) = log₃3 = 1 ✓ <strong>CORRECT</strong></p><p><strong>Option (d):</strong> (2^(1/3) + 5^(1/3))(2^(2/3) - 2^(1/3)·5^(1/3) + 5^(2/3))</p><p>• This matches a³ + b³ = (a+b)(a² - ab + b²) where a = 2^(1/3), b = 5^(1/3)</p><p>• = (2^(1/3))³ + (5^(1/3))³ = 2 + 5 = 7 ✓ <strong>CORRECT</strong></p><p>∴ Answer: B, C, D</p>
Correct Answer: B, C, D