Найдите решение уравнения. Created by Lana2201. algebra-ru

Найдите решение уравнения...

Lana2201 @Lana2201

July 2022 1 4 Report

Найдите решение уравнения

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$log_{1/2}^2(4x)+log_2\frac{x^3}{8}=8\; \; \; ,\; \; \; ODZ:\; x>0\\\\\star \; \; log^2_{1/2}(4x)=\Big (log_{1/2}(4x)\Big )^2=\Big (-log_2(4x)\Big )^2=(-1)^2\cdot \Big (log_24+log_2x\Big )^2=\\\\=(2+log_2x)^2=4+4log_2x+log_2^2x\; ;\\\\\star \; \; log_2\frac{x^3}{8}=log_2(\frac{x}{2})^3=3\, (log_2x-log_22)=3log_2x-3\; ;\\\\4+4\, log_2x+log_2^2x+3log_2x-3=8\\\\log_2^2x+7\, log_2x-7=0\; \; ,\; \; \; D=7^2+4\cdot 7=77\; ,\\\\log_2x=\frac{-7-\sqrt{77}}{2}\; \; ,\; \; \log_2x=\frac{-7+\sqrt{77}}{2}$

$x_1=2^{\frac{-7-\sqrt{77}}{2}}=\sqrt{2^{-7-\sqrt{77}}} \; \; ,\; \; x_2=2^{\frac{-7+\sqrt{77}}{2}}=\sqrt{2^{-7+\sqrt{77}}}$

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