The height of 5-balnced tree is O(logn)
Suppose that the tree has depth $D$. Then all leaves are at distance at least $D/5$ from the root, which means that the tree contains the full binary tree of depth $D/5$ as a subtree. This implies that $n \geq 2^{D/5}$, and so $D = O(\log n)$.

Note that this argument doesn't use the recursive part of the definition.