Is the runtime of binary search big omega of logarithm of n?
First, you can analyze the time complexity of binary search in whatever case you wish, say "best case" and "worst case". In the best case, you use $f(n)$ time, while in the worst case you use $g(n)$ time. These are two functions of $n$, describing two different scenarios you have defined. Now, you can make statements such as $f(n) = O(1)$ or $g(n) = \Theta(\log n)$. However, the Big Oh of a function has nothing to do with time complexity, it is merely something applied to a function that could just as well be representing the number of apples you get for $n$ euros.

In particular, thinking along the lines of "$O$ is always for best case, $\Omega$ is for the worst case" (or whatever), is wrong and dangerous.

To you specific question: if a function $g(n) = \Omega(\log n)$ and $g(n) = O(\log n)$, then $g(n) = \Theta(\log n)$. The function $g(n)$ might be representing the time complexity of binary search or the number of apples, it doesn't matter. And as always, look at the definitions and you can stop guessing.