Finding a local peak in an array in O(log N)?
No, it is not possible to find the max (peak) element in an unsorted array better than $\mathcal{O}(n)$.

When you executed your algorithm $\mathcal{A}$ that has $c \log n$ compare operations, that means; $\mathcal{A}$ only compared $c \log n$ elements. The remaining are not compared by $\mathcal{A}$.

We can easily construct an adversary argument for any $\mathcal{O}(\log n)$ algorithm by just placing the peak, within the elements for $\mathcal{A}$ didn't touch.