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Logarithmic equation for y

CAT 2020 Slot 1 · Quantitative Ability · Medium · Logarithms

This is a medium Quantitative Ability question from the CAT 2020 Slot 1 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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If log⁡45=(log⁡4y)(log⁡65)\log_4 5 = (\log_4 y)(\log_6 \sqrt{5}), then yy equals

(TITA)

Answer

36

Explanation

Given: log⁡45=(log⁡4y)(log⁡651/2)\log_4 5 = (\log_4 y)(\log_6 5^{1/2}) log⁡45=(log⁡4y)⋅12log⁡65\log_4 5 = (\log_4 y) \cdot \frac{1}{2} \log_6 5

Divide both sides by log⁡45\log_4 5: 1=log⁡4ylog⁡45⋅12log⁡651 = \frac{\log_4 y}{\log_4 5} \cdot \frac{1}{2} \log_6 5 1=log⁡5y⋅12log⁡561 = \log_5 y \cdot \frac{1}{2 \log_5 6} 2log⁡56=log⁡5y2 \log_5 6 = \log_5 y log⁡562=log⁡5y  ⟹  y=36\log_5 6^2 = \log_5 y \implies y = 36

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