Bài giải

PT $\Leftrightarrow \left\{ \begin{align}& x>0 \\ & {{x}^{3}}+1>0 \\ & {{x}^{2}}-x+1>0 \\ & {{\log }_{{{2}^{{}}}}}({{x}^{3}}+1)-{{\log }_{2}}({{x}^{2}}-x+1)-2{{\log }_{{{2}^{{}}}}}x=0 \\ \end{align} \right.$

$\Leftrightarrow \left\{ \begin{align}& x>0 \\ & {{\log }_{2}}\frac{{{x}^{3}}+1}{{{x}^{2}}-x+1}-{{\log }_{2}}x=0 \\ \end{align} \right.$

$\Leftrightarrow \left\{ \begin{align}& x>0 \\ & {{\log }_{2}}\frac{{{x}^{3}}+1}{x({{x}^{2}}-x+1)}=0 \\ \end{align} \right.$

$\Leftrightarrow \left\{ \begin{align}& x>0 \\ & \frac{{{x}^{3}}+1}{x({{x}^{2}}-x+1)}=1 \\ \end{align} \right.$

$\Leftrightarrow \left\{ \begin{align}& x>0 \\ & \frac{(x+1)({{x}^{2}}-x+1)}{x({{x}^{2}}-x+1)}=1 \\ \end{align} \right.$

$\Leftrightarrow \left\{ \begin{align}& x>0 \\ & x+1=x \\ \end{align} \right.$ $\Leftrightarrow \left\{ \begin{align}& x>0 \\  & 0.x=-1 \\ \end{align} \right.\Rightarrow x\in \varnothing $