Bounds on logarithmic expression

CAT 2005 Slot 1 · QA · Medium · Algebra

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If x≥yx \ge y and y>1y > 1, then the value of the expression log⁡x(xy)+log⁡y(yx)\log_x\left(\frac{x}{y}\right) + \log_y\left(\frac{y}{x}\right) can never be

  1. A.

    −1-1

  2. B.

    −0.5-0.5

  3. C.

    0

  4. D.

    1

Answer

D

Explanation

Expression E=(log⁡xx−log⁡xy)+(log⁡yy−log⁡yx)=1−log⁡xy+1−log⁡yx=2−(log⁡xy+1log⁡xy)E = (\log_x x - \log_x y) + (\log_y y - \log_y x) = 1 - \log_x y + 1 - \log_y x = 2 - \left(\log_x y + \frac{1}{\log_x y}\right). Let t=log⁡xyt = \log_x y. Since x≥y>1x \ge y > 1, 0<t≤10 < t \le 1. By AM-GM inequality, t+1t≥2t + \frac{1}{t} \ge 2. Thus E=2−(t+1t)≤2−2=0E = 2 - \left(t + \frac{1}{t}\right) \le 2 - 2 = 0. Since E≤0E \le 0, it can never be 11.

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