equação logaritmica

\[\mathrm{[log_2(4)]\cdot 2^x+\left [log_2\left ( 2^{-1} \right )  \right ]\cdot 2^x-[log_{10}(100)]\cdot (2^x)^2=0}\]

\[\mathrm{2\cdot 2^x-2^x-2\cdot (2^x)^2=0\to 2^x-2\cdot (2^x)^2=0\to 2^x(1-2\cdot 2^x)=0}\]

\[\mathrm{2^x=0\ \therefore\  x=0\ e\ 2^x=2^{-1}\ \therefore\ x=-1}\]

\[\mathrm{Testando\ para\ x = 0:[log_2(4)]\cdot 2^0+\left [log_2\left ( 2^{-1} \right )  \right ]\cdot 2^0-[log_{10}(100)]\cdot (2^0)^2=0\to -1=0\ (Falso)}\]

\[\mathrm{Testando\ para\ x = 1:[log_2(4)]\cdot 2^{-1}+\left [log_2\left ( 2^{-1} \right )  \right ]\cdot 2^{-1}-[log_{10}(100)]\cdot (2^{-1})^2=0\to 0=0\ (Ok)}\]

\[\mathrm{\therefore\ S=\left \{ -1 \right \}}\]