B-tree is a data structure, which looks like this:
If I want to look for some specific value in this structure, I need to go through several elements in root to find the right child-node. The I need to go through several elements in the child-node to find its right child-node etc.
The point is, when I have $n$ elements in every node, then I have to go through all of them in the worst case. So, we have $O(n)$ complexity for searching in one node.
Then, we must go through all the levels of the structure, and they're $log_m N$ of them, $m$ being the order of B-tree and $N$ the number of all elements in the tree. So here, we have $O(log N)$ complexity in the worst case.
Putting these information together, we should have $O(n) * O(log n) = O(n * log n)$ complexity.
But the complexity is just $O(log n)$ - why? What am I doing wrong?