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Question Number 150853 by mathdanisur last updated on 15/Aug/21

log_(2021)  (√(x : (√(x : (√(x :..)))))) = 674  find  x=?

$$\mathrm{log}_{\mathrm{2021}} \:\sqrt{\mathrm{x}\::\:\sqrt{\mathrm{x}\::\:\sqrt{\mathrm{x}\::..}}}\:=\:\mathrm{674} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

Commented by liberty last updated on 16/Aug/21

(√(x/( (√(x/( (√(x/( (√(x/⋮))))))))))) = 2021^(674)   ⇔(√(x/(2021^(674) ))) = 2021^(674)   ⇔ (x/(2021^(674) )) = 2021^(1348)   ⇔ x = 2021^(2022)

$$\sqrt{\frac{\mathrm{x}}{\:\sqrt{\frac{\mathrm{x}}{\:\sqrt{\frac{\mathrm{x}}{\:\sqrt{\frac{\mathrm{x}}{\vdots}}}}}}}}\:=\:\mathrm{2021}^{\mathrm{674}} \\ $$$$\Leftrightarrow\sqrt{\frac{\mathrm{x}}{\mathrm{2021}^{\mathrm{674}} }}\:=\:\mathrm{2021}^{\mathrm{674}} \\ $$$$\Leftrightarrow\:\frac{\mathrm{x}}{\mathrm{2021}^{\mathrm{674}} }\:=\:\mathrm{2021}^{\mathrm{1348}} \\ $$$$\Leftrightarrow\:\mathrm{x}\:=\:\mathrm{2021}^{\mathrm{2022}} \\ $$

Commented by mathdanisur last updated on 16/Aug/21

Thank you Ser

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser} \\ $$

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