${\log _2}({3.2^x} - 1) = 2x + 1\Leftrightarrow 3.2^x-1=2^{2x+1}$
$\Leftrightarrow 2.(2^x)^2-3.2^x+1=0$
$\Leftrightarrow\left[\begin{array}{I}
2^x=1\\2^x=\frac{1}{2}\end{array}\right.$
$\Leftrightarrow\left[\begin{array}{I}
x=0\\ x=-1\end{array}\right.$