First slide

Functions (XII)

Question

The domain of the function $f (x) = \log_{3} [- {(\log_{3} x)}^{2} + 5 \log_{3} x - 6] is$

Moderate

A

(0, 9) ∪ (27, ∞)
B

[9, 27]
C

(9, 27)
D

None of these

Solution

f (x) is defined if
$\begin{array}{l} - {(\log_{3} x)}^{2} + 5 \log_{3} x - 6 > 0 and x > 0 \\ \Rightarrow (\log_{3} x - 3) (2 - \log_{3} x) > 0 and x > 0 \\ \Rightarrow (\log_{3} x - 2) (\log_{3} x - 3) < 0 and x > 0 \\ \Rightarrow 2 < \log_{3} x < 3 and x > 0 \\ \Rightarrow 3^{2} < x < 3^{3} \Rightarrow 9 < x < 27 \\ ∴ Domain of f = (9, 27) . \end{array}$

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