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Functions: Domain of Logarithmic Function

XAT 2011 · Quantitative Ability & Data Interpretation · Medium · Functions

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The domain of the function f(x)=log⁡7(log⁡3(log⁡5(20x−x2−91)))f(x) = \log_7(\log_3(\log_5(20x - x^2 - 91))) is:

  1. A.

    (7, 13)

  2. B.

    (8, 12)

  3. C.

    (7, 12)

  4. D.

    (12, 13)

  5. E.

    None of the above

Answer

B

Explanation

For f(x)f(x) to be defined: log⁡3(log⁡5(20x−x2−91))>0  ⟹  log⁡5(20x−x2−91)>30=1  ⟹  20x−x2−91>51=5  ⟹  x2−20x+96<0  ⟹  (x−8)(x−12)<0  ⟹  x∈(8,12)\log_3(\log_5(20x - x^2 - 91)) > 0 \implies \log_5(20x - x^2 - 91) > 3^0 = 1 \implies 20x - x^2 - 91 > 5^1 = 5 \implies x^2 - 20x + 96 < 0 \implies (x - 8)(x - 12) < 0 \implies x \in (8, 12).

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