log-5-5-5-5-5-

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

$$\mathrm{log}_{\mathrm{5}} \sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}…..}}}\:}=\:? \\ $$

Commented by bbbbbbbb last updated on 23/Aug/23

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

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

$$\:{let}\:\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}…}}}\:={x} \\ $$$$\Rightarrow\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}…}}}={x}^{\mathrm{2}} \\ $$$$\Rightarrow{x}^{\mathrm{2}} =\mathrm{5}{x} \\ $$$$\therefore{x}=\mathrm{5}\:\:\left[{x}\neq\mathrm{0}\right] \\ $$$$\:{log}_{\mathrm{5}} \mathrm{5}=\mathrm{1} \\ $$

Answered by MM42 last updated on 23/Aug/23

$$\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}…}}}=\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}+…} =\mathrm{5}^{\frac{\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}}} \:=\mathrm{5} \\ $$$$\Rightarrow{log}_{\mathrm{5}} \:\mathrm{5}=\mathrm{1} \\ $$

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