Basic Mathematics & Logarithm
Properties of Logarithms
Grade Class 11
Question:
<p>Which of the following statements is(are) correct?</p>
\(7^{1/7} > 42^{1/14} > 1\)
\(\log_{3} 5 \cdot \log_{7} 9 \cdot \log_{11} 13 > -2\)
sqrt(99 + 70 sqrt2) + sqrt(99 - 70 sqrt2) is rational
\(1/\log_{4} 3 + 1/\log_{7} 3 > 3\)
Step-by-Step Solution
Key Concept: Handle each statement independently. Some require estimation, while others collapse after a clever algebraic simplification.
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Check each statement independently by estimation or algebraic simplification. A is true by raising both sides to the 14th power. B is true because the product is positive, hence certainly greater than -2. C is false since the sum simplifies to 14 sqrt2. D is true because 1/log_4 3 + 1/log_7 3 = log_3 4 + log_3 7 = log_3 28 > 3. So the correct set is A, B and D. Trap: Option B looks suspicious, but because all logs involved are positive the inequality is immediate. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: A, B, D