Question Number 150853 by mathdanisur last updated on 15/Aug/21
$$\mathrm{log}_{\mathrm{2021}} \:\sqrt{\mathrm{x}\::\:\sqrt{\mathrm{x}\::\:\sqrt{\mathrm{x}\::..}}}\:=\:\mathrm{674} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$
Commented by liberty last updated on 16/Aug/21
$$\sqrt{\frac{\mathrm{x}}{\:\sqrt{\frac{\mathrm{x}}{\:\sqrt{\frac{\mathrm{x}}{\:\sqrt{\frac{\mathrm{x}}{\vdots}}}}}}}}\:=\:\mathrm{2021}^{\mathrm{674}} \\ $$$$\Leftrightarrow\sqrt{\frac{\mathrm{x}}{\mathrm{2021}^{\mathrm{674}} }}\:=\:\mathrm{2021}^{\mathrm{674}} \\ $$$$\Leftrightarrow\:\frac{\mathrm{x}}{\mathrm{2021}^{\mathrm{674}} }\:=\:\mathrm{2021}^{\mathrm{1348}} \\ $$$$\Leftrightarrow\:\mathrm{x}\:=\:\mathrm{2021}^{\mathrm{2022}} \\ $$
Commented by mathdanisur last updated on 16/Aug/21
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser} \\ $$