What is the domain of the function f (x) = log4 (log5 (log3 (18x – x

1.	What is the domain of the function f (x) = log4 (log5 (log3 (18x – x2 – 77))).
Answer» log_ba is defined for all a > 0, b > 0 and b ≠ 1 ∴ f (x) will be defined if log₅ (log₃ (18x – x² – 77)) > 0 ⇒ log₃ (18x – x² – 77) > 5⁰, i.e., 1 ⇒ 18x – x² – 77 > 3¹ ⇒ 18x – x² – 80 > 0 ⇒ x² – 18x + 80 < 0 ⇒ (x – 8) (x – 10) < 0 ⇒ 8 < x < 10 ⇒ x ∈ (8, 10). ∴ Domain of f (x) is x ∈ (8, 10).

What is the domain of the function f (x) = log4 (log5 (log3 (18x – x2 – 77))).

Answer»

log_ba is defined for all a > 0, b > 0 and b ≠ 1

∴ f (x) will be defined if log₅ (log₃ (18x – x² – 77)) > 0

⇒ log₃ (18x – x² – 77) > 5⁰, i.e., 1

⇒ 18x – x² – 77 > 3¹

⇒ 18x – x² – 80 > 0

⇒ x² – 18x + 80 < 0

⇒ (x – 8) (x – 10) < 0

⇒ 8 < x < 10

⇒ x ∈ (8, 10).

∴ Domain of f (x) is x ∈ (8, 10).