Why is big omega of peak finding Omega(lg n)?
I think you got a bit confused with the definition of Big-Omega ($\Omega$). 
The $\Omega(f(n))$ is a lower bound for an algorithm. It means that your algorithm's running time can not be less than $kf(n)$ where $k$ is a constant. And it applies to worst case not the best case. 

edit
If you calculate your algorithm running time as $c\log{n}$, it's asymptotic analysis says that it is $\Omega(\log{n})$. 



big omega notation