User:Lmov/Sandbox

$\log _{2x}{\frac {2}{x}}\log _{2}^{2}x+\log _{2}^{4}x=1$

$x\in \left(0;{\frac {1}{2}}\right)\cup \left({\frac {1}{2}};+\infty \right)$

${\frac {\ln {\frac {2}{x}}\ln ^{2}x}{\ln {2x}\ln ^{2}2}}+{\frac {\ln ^{4}x}{\ln ^{4}2}}=1$

$\ln {\frac {2}{x}}\ln ^{2}x\ln ^{2}2+\ln ^{4}x\ln {2x}=\ln {2x}\ln ^{4}2$

$\left(\ln 2-\ln x\right)\ln ^{2}x\ln ^{2}2+\ln ^{4}x\left(\ln 2+\ln x\right)-\left(\ln 2+\ln x\right)\ln ^{4}2=0$

$\ln ^{5}x+\ln ^{4}x\ln 2-\ln ^{3}x\ln ^{2}2+\ln ^{2}x\ln ^{3}2-\ln x\ln ^{4}2-\ln ^{5}2=0$

$\left({\frac {\ln x}{\ln 2}}\right)^{5}+\left({\frac {\ln x}{\ln 2}}\right)^{4}-\left({\frac {\ln x}{\ln 2}}\right)^{3}+\left({\frac {\ln x}{\ln 2}}\right)^{2}-\left({\frac {\ln x}{\ln 2}}\right)-1=0$

$\log _{2}^{5}x+\log _{2}^{4}x-\log _{2}^{3}x+\log _{2}^{2}x-\log _{2}x-1=0$

Using Ruffini's rule find that 1 is a root:

${\begin{matrix}&1&1&-1&1&-1&-1\\1&\left[1\right.&2&1&2&\left.1\right]&0\end{matrix}}$

$\left(\log _{2}x-1\right)\left(\log _{2}^{4}x+2\log _{2}^{3}x+\log _{2}^{2}x+2\log _{2}x+1\right)=0$

$\log _{2}x=1$

$x_{1}=2{\mbox{ (1)}}$

$\log _{2}^{4}x+2\log _{2}^{3}x+\log _{2}^{2}x+2\log _{2}x+1=0$

$\log _{2}^{2}x+2\log _{2}x+1+2{\frac {1}{\log _{2}x}}+{\frac {1}{\log _{2}^{2}x}}=0$

$\left(\log _{2}^{2}x+{\frac {1}{\log _{2}^{2}x}}\right)+2\left(\log _{2}x+{\frac {1}{\log _{2}x}}\right)+1=0$

${\mbox{Let }}y=\log _{2}x+{\frac {1}{\log _{2}x}}\Rightarrow y^{2}-2=\log _{2}^{2}x+{\frac {1}{\log _{2}^{2}x}}$

$y^{2}-2+2y+1=0$

$y^{2}+2y-1=0$

$y_{1,2}=-1\pm {\sqrt {2}}$

$\log _{2}x+{\frac {1}{\log _{2}x}}=-1-{\sqrt {2}}$

$\log _{2}^{2}x+\left({\sqrt {2}}+1\right)\log _{2}x+1=0$

$\log _{2}x={\frac {-\left({\sqrt {2}}+1\right)\pm {\sqrt {2{\sqrt {2}}-1}}}{2}}$

$x_{2,3}=2^{\frac {-\left({\sqrt {2}}+1\right)\pm {\sqrt {2{\sqrt {2}}-1}}}{2}}{\mbox{ (2)}}$

$\log _{2}x+{\frac {1}{\log _{2}x}}=-1+{\sqrt {2}}$

$\log _{2}^{2}x+\left(1-{\sqrt {2}}\right)\log _{2}x+1=0$

$D=\left(1-{\sqrt {2}}\right)^{2}-4=-1-2{\sqrt {2}}<0\Rightarrow {\mbox{roots are complex}}$

Thus, from (1) and (2):

$x_{1}=2,x_{2,3}=2^{\frac {-\left({\sqrt {2}}+1\right)\pm {\sqrt {2{\sqrt {2}}-1}}}{2}}$