Usually, a logarithmic time means O(N log N), which is worse than O(N) but much better than O(N²). For example, the best sorting algorithms have O(N log N) times (Quicksort with possible degeneration, Heap Sort), while naive ones have O(N²) times.

As for your algorithm:
– You don’t make the difference between strictly increasing (resp. strictly decreasing) and simply non-decreasing (resp. non-increasing).
– AFAIK, it is only valid for arrays indexed by consecutive integers. A foreach would be better suited, keeping the previous element in a temporary variable (which could be more efficient than an array access as PHP does copy-on-write. This should be tested).
– I’d rather use multiple functions than a single parametrized one. This eliminates the need for a constant and limitates the needed logic.

Cheers.