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System of Logarithmic Equations

CAT 2019 Slot 1 · Quantitative Ability · Medium · Logarithms

This is a medium Quantitative Ability question from the CAT 2019 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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Let xx and yy be positive real numbers such that log⁡5(x+y)+log⁡5(x−y)=3\log_5(x + y) + \log_5(x - y) = 3, and log⁡2y−log⁡2x=1−log⁡23\log_2 y - \log_2 x = 1 - \log_2 3. Then xyxy equals

  1. A.

    25

  2. B.

    150

  3. C.

    250

  4. D.

    100

Answer

B

Explanation

First equation: log⁡5((x+y)(x−y))=3  ⟹  x2−y2=53=125\log_5((x+y)(x-y)) = 3 \implies x^2 - y^2 = 5^3 = 125. Second equation: log⁡2(y/x)=log⁡2(2/3)  ⟹  yx=23  ⟹  y=23x\log_2(y/x) = \log_2(2/3) \implies \frac{y}{x} = \frac{2}{3} \implies y = \frac{2}{3}x. Substitute into first equation: x2−49x2=125  ⟹  59x2=125  ⟹  x2=225  ⟹  x=15x^2 - \frac{4}{9}x^2 = 125 \implies \frac{5}{9}x^2 = 125 \implies x^2 = 225 \implies x = 15. y=23(15)=10y = \frac{2}{3}(15) = 10. xy=15×10=150xy = 15 \times 10 = 150.

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