linear algorithm

The OP code does not complete in \$O(\log n)\$ logarithmic time.

In general a correct algorithm will have at least \$O(n)\$ linear complexity. It cannot do better than that, since the maximum element could be hiding at any position.

code

This line of source code is correct:

    int max = arr[0];

In particular, initializing in this way is better than using a \$- \infty\$ sentinel such as \$-(2^{63})\$.

The remainder of the code should be rewritten, as a short loop.

(When initially reading it, I was kind of hoping that assignment would immediately produce the answer in \$O(1)\$ constant time by virtue of input being in monotonic decreasing sorted order, but then I saw the problem puts no constraint on the input vector.)

Please delete that "Write your code here" comment line.

Just write a linear scan of the input integers, and you're done. No need to approach it from both ends -- ascending indexes will suffice.

braces

        if(arr[i]>=max  && arr[i] >= arr[j]) max = arr[i];

Please don't do that. Surround if / else statements with { } braces, even if there's only a single statement as we see here.

The way the OP code is phrased, it invites maintenance headaches down the road. We don't want some hapless engineer to add a second statement and mispredict how the compiled code will behave.