Maths · Functions
The domain of the function = \) \right) \) is \right.
The domain of the function \( \boldsymbol{f}(\boldsymbol{x})= \) \( \log _{4}\left(\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right) \) is \right.
- A. \( x \in(4,5) \)
- B. \( x \in(0,10) \)
- C. \( x \in(8,10) \)
- D. None of these
Step-by-step solution
The innermost log requires 18x - x^2 - 77 > 0, giving x ∈ (7,11). Next, log_3(...) > 0 gives 18x - x^2 - 77 > 1, yielding x ∈ (9-√3, 9+√3) ≈ (7.268, 10.732). Finally, log_5(...) > 0 gives log_3(...) > 1, so 18x - x^2 - 77 > 3, leading to x ∈ (8,10). Intersection of all conditions gives x ∈ (8,10).
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