[tex]log_2 (2^{-x+1}+1)=x \Rightarrow 2^{-x+1}+1 = 2^x \Rightarrow2^{-x}\cdot2-2^x+1 = 0 \Rightarrow \\ \Rightarrow \frac{1}{2^x}\cdot2-2^x+1=0 \\ \\ Not:2^x=t, \quad t\ \textgreater \ 0 \\ \\ \Rightarrow \frac{1}{t}\cdot2-t+1=0 \Rightarrow 2-t^2+t = 0 \Rightarrow -t^2+t+2=0\Rightarrow \\ \\ \Rightarrow t^2-t-2=0 \\ \Delta = 1+8 = 9 \\ \\ t_{1,2} = \frac{1\pm3}{2} \Rightarrow \left \{ {{t_1=2} \atop {t_2=-1}(F)} \right. \\ \\ 2^x = 2 \Rightarrow x = 1 \Rightarrow S=\{1\}[/tex]
