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Question Number 164985 by cortano1 last updated on 24/Jan/22

Answered by Eulerian last updated on 24/Jan/22

$$$$$$\mathbf{Solution}:$$$$3\cdot \frac{{\log }^2\left(\mathrm{x}\right)}{{\log }^2\left(2\right)}-1-9\cdot \frac{{\log }^2\left(2\right)}{{\log }^2\left(\mathrm{x}\right)}=25$$$$$$$$\mathrm{let}:\mathrm{k}=\frac{{\log }^2\left(\mathrm{x}\right)}{{\log }^2\left(2\right)}$$$$\therefore$$$$3\mathrm{k}-\frac{9}{\mathrm{k}}=26$$$${3\mathrm{k}}^2-26\mathrm{k}-9=0$$$$$$$$\mathrm{By}\mathrm{quadratic}\mathrm{formula}:$$$$\mathrm{k}=\frac{26\pm \sqrt{{26}^2-4\left(3\right)(-9)}}{2\left(3\right)}=\frac{26\pm 28}{6}$$$$\therefore$$$${\mathrm{k}}_1=9$$$${\mathrm{k}}_2=-\frac{1}{3}$$$$$$$$\mathrm{By}\mathrm{substituting}\mathrm{back}:$$$${\log }^2\left(\mathrm{x}\right)=9\cdot {\log }^2\left(2\right)$$$$\log \left(\mathrm{x}\right)=3\cdot \log \left(2\right)$$$$\mathrm{x}=8$$$$$$$$\mathrm{and}$$$$$$$$\log \left(\mathrm{x}\right)=\frac{\mathrm{i}\sqrt{3}}{3}\cdot \log \left(2\right)$$$$\mathrm{x}=2^{\frac{\mathrm{i}\sqrt{3}}{3}}$$$$$$

Answered by alephzero last updated on 24/Jan/22

$$\sqrt{{3\log }_2^{2}x-1-{9\log }_x^{2}2}=5$$$${3\log }_2^{2}x-{9\log }_x^{2}2-1=25$$$${3\log }_2^{2}x-\frac{9}{{\log }_2^{2}x}=26$$$$\mathrm{Let}t={\log }_2^{2}x$$$$\Rightarrow 3t-\frac{9}{t}=26$$$$3t-\frac{9}{t}-26=0$$$$3t^2-26t-9=0$$$$t=-\frac{1}{3},9$$$$\{\begin{array}{l}{\log }_2^{2}x=-\frac{1}{3}\\ {\log }_2^{2}x=9\end{array}$$$$\{\begin{array}{l}x\notin \mathbb{R}\\ x=\frac{1}{8},8\end{array}$$$$\Rightarrow x=\frac{1}{8},8$$

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