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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

1 Answer

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by (24.0k points)
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Best answer

(b) 9.

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

⇒ log3x + log3x1/2 + log3x1/4 + log3 (x1/8) + ..... ∞ = 4 

⇒ log3 (x . x1/2 . x1/4 . x1/8 ..... ∞) = 4 

⇒ log3 (x1 + 1/2 + 1/4 + 1/8 ..... ∞) = 4

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

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

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