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Logarithmic equation range for a

CAT 2021 Slot 3 · Quantitative Ability · Easy · Logarithms

This is an easy Quantitative Ability question from the CAT 2021 Slot 3 paper. It tests Logarithms. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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For a real number aa, if log⁡15a+log⁡32a(log⁡15a)(log⁡32a)=4\frac{\log_{15} a + \log_{32} a}{(\log_{15} a)(\log_{32} a)} = 4 then aa 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

Explanation

We can rewrite the given expression using base-change formula: log⁡15a+log⁡32a(log⁡15a)(log⁡32a)=1log⁡a15+1log⁡a32=log⁡a15+log⁡a32=log⁡a(15×32)=log⁡a480\frac{\log_{15} a + \log_{32} a}{(\log_{15} a)(\log_{32} a)} = \frac{1}{\log_a 15} + \frac{1}{\log_a 32} = \log_a 15 + \log_a 32 = \log_a (15 \times 32) = \log_a 480

Given log⁡a480=4  ⟹  a4=480\log_a 480 = 4 \implies a^4 = 480.

Since 44=2564^4 = 256 and 54=6255^4 = 625, and 256<480<625256 < 480 < 625, we get: 4<a<54 < a < 5.

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