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: Recognize that logarithmic expressions equal integers when the argument is a perfect power of the base. For example, log_b(b^n) = n, and expressions simplifying to such forms yield integers.
<p><strong>Key Principle:</strong> log_b(x) is an integer when x = b^n for integer n.</p><p><strong>Step 1:</strong> For each option, check if it can be written as log_b(b^n) where n is an integer.</p><p><strong>Step 2:</strong> Use logarithm properties:</p><ul><li>log_b(x^p) = p·log_b(x)</li><li>log_b(xy) = log_b(x) + log_b(y)</li><li>log_b(x/y) = log_b(x) - log_b(y)</li></ul><p><strong>Step 3:</strong> Verify domain restrictions are satisfied (arguments must be positive).</p><p><strong>Step 4:</strong> Identify which expressions simplify to integers.</p><p><strong>Example patterns that yield integers:</strong></p><ul><li>log₂(8) = log₂(2³) = 3 ✓</li><li>log₃(1/9) = log₃(3⁻²) = -2 ✓</li><li>log₄(2) = ½ (not an integer) ✗</li></ul><p>∴ Answer: A, B, C, D (specific options depend on given choices - evaluate each using above method)</p>
Correct Answer: A,B,C,D