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Question Number 196355 by MATHEMATICSAM last updated on 23/Aug/23

\log_{5} \sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \dots . .}}}} = ?

Commented by bbbbbbbb last updated on 23/Aug/23

\log_{5} \sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \dots . .}}}} = ?

\sqrt{x \sqrt{x \sqrt{x \sqrt{x \dots . .}}}} = x

\log_{5} \sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \dots}}}} = \log_{5} 5 = 1

Answered by som(math1967) last updated on 23/Aug/23

l e t \sqrt{5 \sqrt{5 \sqrt{5 \dots}}} = x

\Rightarrow 5 \sqrt{5 \sqrt{5 \sqrt{5 \dots}}} = x^{2}

\Rightarrow x^{2} = 5 x

∴ x = 5 [x \neq 0]

{l o g}_{5} 5 = 1

Answered by MM42 last updated on 23/Aug/23

\sqrt{5 \sqrt{5 \sqrt{5 \dots}}} = 5^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots} = 5^{\frac{\frac{1}{2}}{1 - \frac{1}{2}}} = 5

\Rightarrow {l o g}_{5} 5 = 1

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