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

$${\log }_5\sqrt{5\sqrt{5\sqrt{5\sqrt{5.....}}}}=?$$

Commented by bbbbbbbb last updated on 23/Aug/23

$${\log }_5\sqrt{5\sqrt{5\sqrt{5\sqrt{5.....}}}}=?$$$$\sqrt{\mathrm{x}\sqrt{\mathrm{x}\sqrt{\mathrm{x}\sqrt{\mathrm{x}.....}}}}=\mathrm{x}$$$${\log }_5\sqrt{5\sqrt{5\sqrt{5\sqrt{5...}}}}={\log }_{5}5=1$$

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

$$let\sqrt{5\sqrt{5\sqrt{5...}}}=x$$$$\Rightarrow 5\sqrt{5\sqrt{5\sqrt{5...}}}=x^2$$$$\Rightarrow x^2=5x$$$$\therefore x=5[x\ne 0]$$$${log}_{5}5=1$$

Answered by MM42 last updated on 23/Aug/23

$$\sqrt{5\sqrt{5\sqrt{5...}}}=5^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...}=5^{\frac{\frac{1}{2}}{1-\frac{1}{2}}}=5$$$$\Rightarrow {log}_{5}5=1$$

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