The number of meaningful solutions of log4(x – 1) = log2 (x – 3) i

1.	The number of meaningful solutions of log4(x – 1) = log2 (x – 3) is(a) zero (b) 1 (c) 2 (d) 3
Answer» (b) 1 log₄(x – 1) = log₂(x – 3) ⇒ log_2² (x − 1) = log₂(x – 3) ⇒ \(\frac{1}{2}\) log₂ (x–1) = log₂ (x– 3) ⇒ log₂ (x–1) = 2 log₂ (x– 3) \(\big[\)Using log_a^m(bⁿ) = \(\frac{n}{m}\) log_a b\(\big]\) ⇒ log₂(x – 1) = log₂(x – 3)² ⇒ (x – 1) = (x – 3)² ⇒ x – 1 = x² – 6x + 9 ⇒ x² – 7x + 10 = 0 ⇒ (x – 2) (x – 5) = 0 ⇒ x = 2 or 5 Neglecting x = 2 as log₂(x – 3) is defined when x > 2. ⇒ There is only one meaningful solution of the given equation.

The number of meaningful solutions of log4(x – 1) = log2 (x – 3) is(a) zero (b) 1 (c) 2 (d) 3

Answer»

(b) 1

log₄(x – 1) = log₂(x – 3) ⇒ log_2² (x − 1) = log₂(x – 3)

⇒ \(\frac{1}{2}\) log₂ (x–1) = log₂ (x– 3) ⇒ log₂ (x–1) = 2 log₂ (x– 3)

\(\big[\)Using log_a^m(bⁿ) = \(\frac{n}{m}\) log_a b\(\big]\)

⇒ log₂(x – 1) = log₂(x – 3)²

⇒ (x – 1) = (x – 3)² ⇒ x – 1 = x² – 6x + 9

⇒ x² – 7x + 10 = 0 ⇒ (x – 2) (x – 5) = 0 ⇒ x = 2 or 5

Neglecting x = 2 as log₂(x – 3) is defined when x > 2.

⇒ There is only one meaningful solution of the given equation.