Basic Mathematics & Logarithm
Change of Base Formula
Grade Class 11
Question:
<p>Which of the following statements are true?</p>
\(\log_{2} 3 < \log_{12} 10\)
\(\log_{6} 5 < \log_{7} 8\)
\(\log_{3} 26 < \log_{2} 9\)
\(\log_{16} 15 > \log_{10} 11 > \log_{7} 6\)
Step-by-Step Solution
Key Concept: Convert all logarithms to one common base when comparing sizes. That keeps the inequalities honest.
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. Compare by converting to natural logs or using monotonicity. Direct comparison shows B and C are true while A and D fail under common-log estimates. Hence the true statements are B and C. Trap: Comparisons of logarithms with different bases are usually easiest after converting all of them to one base. 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: B, C