FullSimplify understands how to reduce the root:
FullSimplify[Sqrt[3 - 2 Sqrt[2]] (1 + Sqrt[2])]
(* 1 *)
However, it apparently starts down a different path and never tries this. Reduce the root first, and then simplify:
FullSimplify[RootReduce[Log[Sqrt[3 - 2 Sqrt[2]] (1 + Sqrt[2])]]]
(* 0 *)
