1. **State the problem:** We have the function $$h(x) = 5 - 2 \cdot 3^{7-x}$$ and its inverse function $$k(x) = h^{-1}(x)$$. We want to find the value of $$x$$ such that $$k(x) = 5$$. 2. **Recall the property of inverse functions:** If $$k = h^{-1}$$, then $$k(x) = y$$ means $$h(y) = x$$. So, $$k(x) = 5$$ implies $$h(5) = x$$. 3. **Calculate $$h(5)$$:** $$ h(5) = 5 - 2 \cdot 3^{7-5} = 5 - 2 \cdot 3^2 = 5 - 2 \cdot 9 = 5 - 18 = -13 $$ 4. **Conclusion:** Since $$h(5) = -13$$, then $$k(-13) = 5$$. The problem asks for the value of $$x$$ such that $$k(x) = 5$$, so the answer is $$x = -13$$. **Final answer:** $$\boxed{-13}$$