By using a suitable substitution, solve the equation $e^{2x} = 4(1 - e^{-2x})$.

1. Introduce a substitution to transform the equation into a polynomial

Let $u = e^{2x}$. Then $e^{-2x} = \frac{1}{u}$.
The equation becomes:
$u = 4\left(1 - \frac{1}{u}\right)$

2. Rearrange into a quadratic equation and solve for $u$

$u = 4 - \frac{4}{u}$
Multiply by $u$:
$u^2 = 4u - 4$
$u^2 - 4u + 4 = 0$
$(u - 2)^2 = 0$
$u = 2$

3. Substitute back and solve for $x$

$e^{2x} = 2$
$2x = \ln 2$
$x = \frac{1}{2} \ln 2$