Logarithm Product Identity

CAT 2006 Slot 1 · QA · Medium · Quantitative Aptitude

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If log⁡yx=a⋅log⁡zy=b⋅log⁡xz=ab\log_y x = a \cdot \log_z y = b \cdot \log_x z = ab, then which of the following pairs of values for (a,b)(a, b) is not possible?

  1. A.

    (-2, 1/2)

  2. B.

    (1, 1)

  3. C.

    (0.4, 2.5)

  4. D.

    (pi, 1/pi)

  5. E.

    (2, 2)

Answer

(2, 2)

Explanation

From the given equations: log⁡yx=ab\log_y x = ab log⁡zy=b\log_z y = b log⁡xz=a\log_x z = a

Multiply the three logarithms: (log⁡yx)⋅(log⁡zy)⋅(log⁡xz)=1(\log_y x) \cdot (\log_z y) \cdot (\log_x z) = 1. Substituting their values: (ab)⋅b⋅a=1  ⟹  a2b2=1  ⟹  ab=±1(ab) \cdot b \cdot a = 1 \implies a^2 b^2 = 1 \implies ab = \pm 1. Check options for ab=±1ab = \pm 1:

  1. (−2)×(1/2)=−1(-2) \times (1/2) = -1 (Possible)
  2. 1×1=11 \times 1 = 1 (Possible)
  3. 0.4×2.5=10.4 \times 2.5 = 1 (Possible)
  4. π×(1/π)=1\pi \times (1/\pi) = 1 (Possible)
  5. 2×2=4≠±12 \times 2 = 4 \ne \pm 1 (NOT possible).

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