If the elements need not be distinct, then you cannot have an $O(\log n)$ time algorithm.

Consider the sorted array $[0,0, \dots, 1]$ which has been cyclic shifted $k$ (unknown) times and you need to find where the $1$ appears. This needs $\Omega(n)$ time, as you need to examine at least $n-1$ elements.

However, if you assume the elements are distinct, then you can indeed give an $O(\log n)$ time algorithm.

Assume the array was sorted ascending. Once it is cyclic shifted, we will have that, in the rotated array (say $a[1,2, \dots n]$), that $a[1] \gt a[n]$. (It might help to draw a figure here).

Now if you pick a $j$, you compare $a[j]$ with $a[1]$ and move right or left, depending on whether it is greater or lesser, like binary search.

If the elements need not be distinct, then you cannot have an $O(\log n)$ time algorithm.

Consider the sorted array $[0,0, \dots, 1]$ which has been cyclic shifted $k$ (unknown) times and you need to find where the $1$ appears. This needs $\Omega(n)$ time, as you need to examine at least $n-1$ elements.

However, if you assume the elements are distinct, then you can indeed give an $O(\log n)$ time algorithm.

Assume the array was sorted ascending. Once it is cyclic shifted, we will have that, in the rotated array (say $a[1,2, \dots n]$), that $a[1] \gt a[n]$. (It might help to draw a figure here, plotting $i$ on x-axis and $a[i]$ on the y-axis).

Now if you pick a $j$, you compare $a[j]$ with $a[1]$ and move right or left, depending on whether it is greater or lesser, like binary search.