Question

The number of solutions satisfying the given equation \(x^{\big[(log_3\,x)^2-\frac{9}{2}log_3\,x+5\big]}\) = 3√3 for x ∈ R are :

(a) 0 

(b) 1 

(c) 2 

(d) 3

Answer 1

(d) 3

Taking log of both the sides to base 3, we have,

\(\big[(log_3\,x)^2-\frac{9}{2}log_3\,x+5\big]\) log₃x = log₃3^3/2 = \(\frac{3}{2}\)        (∵ log₃3 = 1)

⇒ 2(log₃x)³ – 9(log₃x)² + 10 log₃x – 3 = 0 

⇒ 2y³ – 9y² + 10y – 3 = 0               (Take log₃x = y)

⇒ (y – 1) (y – 3) (2y – 1) = 0          (Factorising) 

⇒ (log₃x– 1) (log₃x – 3) (2 log₃x – 1) = 0 

⇒ log₃x = 1, log₃x = 3, 2 log₃x = 1 ⇒ x = 3¹, x = 3³, x² = 3¹

⇒ \(x\) = (3, 27, √3)

∴ There are three solutions.