Linear Intersecting Exponential

Algebra Level 2

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 \(3^{\alpha}?\)

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