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Domain for real logarithmic square root

CAT 2019 Slot 2 · Quantitative Ability · Hard · Logarithms

This is a hard Quantitative Ability question from the CAT 2019 Slot 2 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 xx is a real number, then log⁡e(4x−x23)\sqrt{\log_e \left(\frac{4x-x^2}{3}\right)} is a real number if and only if

  1. A.

    −3≤x≤3-3 \le x \le 3

  2. B.

    1≤x≤21 \le x \le 2

  3. C.

    1≤x≤31 \le x \le 3

  4. D.

    −1≤x≤3-1 \le x \le 3

Answer

C

Explanation

For the square root of a logarithm to be real, the quantity inside the square root must be non-negative: log⁡e(4x−x23)≥0\log_e \left(\frac{4x-x^2}{3}\right) \ge 0 4x−x23≥e0=1\frac{4x-x^2}{3} \ge e^0 = 1 4x−x2≥34x - x^2 \ge 3 x2−4x+3≤0x^2 - 4x + 3 \le 0 (x−1)(x−3)≤0  ⟹  1≤x≤3(x - 1)(x - 3) \le 0 \implies 1 \le x \le 3

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