CAT 2021 Slot 3 QA Q6 Logarithms & Exponents

Question 6 of the Quantitative Ability (QA) section, CAT 2021 Slot 3, with CAT's official answer and a worked solution.

For a real number a, if \(\frac{\log _{15} a+\log _{32} a}{\left(\log _{15} a\right)\left(\log _{32} a\right)}=\) = 4 then a must lie in the range
  1. A
    4 < a < 5
  2. B
    3 < a < 4
  3. C
    a > 5
  4. D
    2 < a < 3

Answer: (A) 4 < a < 5 Logarithms & Exponents MCQ

Solution

  1. Divide: \(\frac{1}{\log_{32} a} + \frac{1}{\log_{15} a} = 4\), i.e. \(\log_a 32 + \log_a 15 = 4\).
  2. \(\log_a 480 = 4 \Rightarrow a = 480^{1/4}\).
  3. \(4^4 = 256 \lt 480 \lt 625 = 5^4\), so \(4 \lt a \lt 5\).

Same concept in other years

See every question on Change of base: log_a b = 1 / log_b a.

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