If x > 0 and log3x + log3 \(\sqrt{x}\) + \(\text{log}_3 (\sqrt[4]x)\

1.	If x > 0 and log3x + log3 \(\sqrt{x}\) + \(\text{log}_3 (\sqrt[4]x)\) + \(\text{log}_3 (\sqrt[8]x)\)+ \(\text{log}_3 (\sqrt[16]x)\)+ ..... = 4, then x equals(a) 1 (b) 9 (c) 27 (d) 81
Answer» (b) 9. log₃x + log₃ \(\sqrt{x}\) + \(\text{log}_3 (\sqrt[4]x)\) + \(\text{log}_3 (\sqrt[8]x)\)+ ..... \(\infty\) = 4 ⇒ log₃x + log₃x^1/2 + log₃x^1/4+ log₃ (x^1/8) + ..... ∞ = 4 ⇒ log₃ (x . x^1/2 . x^1/4 . x^1/8 ..... ∞) = 4 ⇒ log₃ (x^{1 + 1/2 + 1/4 + 1/8} ..... ∞) = 4 ⇒ log₃\(x^{\frac{1}{1-\frac{1}{2}}}\) = 4 (∞ S_∞ = \(\frac{a}{r}\) – r) ⇒ log₃ x² = 4 ⇒ x² = 3⁴ = (3²)² ⇒ \(x\) = 9.

If x > 0 and log3x + log3 \(\sqrt{x}\) + \(\text{log}_3 (\sqrt[4]x)\) + \(\text{log}_3 (\sqrt[8]x)\)+ \(\text{log}_3 (\sqrt[16]x)\)+ ..... = 4, then x equals(a) 1 (b) 9 (c) 27 (d) 81

Answer»

(b) 9.

log₃x + log₃ \(\sqrt{x}\) + \(\text{log}_3 (\sqrt[4]x)\) + \(\text{log}_3 (\sqrt[8]x)\)+ ..... \(\infty\) = 4

⇒ log₃x + log₃x^1/2 + log₃x^1/4+ log₃ (x^1/8) + ..... ∞ = 4

⇒ log₃ (x . x^1/2 . x^1/4 . x^1/8 ..... ∞) = 4

⇒ log₃ (x^{1 + 1/2 + 1/4 + 1/8} ..... ∞) = 4

⇒ log₃\(x^{\frac{1}{1-\frac{1}{2}}}\) = 4 (∞ S_∞ = \(\frac{a}{r}\) – r)

⇒ log₃ x² = 4 ⇒ x² = 3⁴ = (3²)² ⇒ \(x\) = 9.