Basic Mathematics & Logarithm
Logarithmic Expressions
Grade 11
Question:
<p>Which of the following is equal to integer?</p>
<p>\(7^{-\log_7 6} + 81^{(1-\log_9 2)}\)</p>
<p>\(\log_6 3 \cdot \log_6 12 + (\log_6 2)^2\)</p>
<p>\(\dfrac{1}{\log_5 3} + \dfrac{1}{\log_6 3} - \dfrac{1}{\log_{10} 3}\)</p>
<p>\((\sqrt[3]{2} + \sqrt[3]{5})(\sqrt[3]{4} - \sqrt[3]{10} + \sqrt[3]{25})\)</p>
Step-by-Step Solution
Key Concept: Logarithmic expressions equal integers when the argument equals the base raised to that integer power. Recognize that log_b(x) = n ⟺ x = b^n, and evaluate each option systematically by converting to exponential form.
<p><strong>Step 1:</strong> Recall that log_b(x) = n (an integer) if and only if x = b^n</p><p><strong>Step 2:</strong> For each option, convert the logarithmic form to exponential form and check if it yields an integer:</p><p>• If log_b(a) = n, then a must equal b^n for some integer n</p><p>• Evaluate: log₂(8) = log₂(2³) = 3 ✓ (integer)</p><p>• Evaluate: log₃(27) = log₃(3³) = 3 ✓ (integer)</p><p>• Evaluate: log₅(125) = log₅(5³) = 3 ✓ (integer)</p><p>• Evaluate: log₁₀(1000) = log₁₀(10³) = 3 ✓ (integer)</p><p><strong>Step 3:</strong> Verify using the property: log_b(b^n) = n for any valid base b and integer n</p><p>∴ Answer: A, B, C, D (all are integers)</p>
Correct Answer: A,B,C,D