Linear Intersecting Exponential

Let $A$ and $B$ the two intersection points of the curve $y=2 \cdot 3^x$ and a line with slope $1$ . Let $\alpha$ and $\beta$ $(\alpha < \beta$ ) be the $x$ -coordinates of $A$ and $B,$ respectively. If the length of line segment $AB$ is $4\sqrt{2}$ , what is the value of $3^{\alpha}?$