Proving that $\text{PCP}(O(\log n),1)\subseteq \mathsf{P}$
By the basic equivalence of PCPs and MaxCSP problems, there exists a conjunction $C$ of polynomially many boolean literals, s.t. 


  $\exists \text{ proof } \pi$ which makes the verifier $V$ accept with probability $\geq p$ $\Leftrightarrow$ $\exists$ an assignment that satisfies at least a $p$ fraction of the literals in $C$.


So you just need to show that the following problem is polytime solvable: given a conjunction $C$ of boolean literals, what is the largest number of literals that can be simultaneously satisfied. That's a very easy problem. 

The idea here is that a PCP with query complexity $q$ corresponds to some MaxCSP problem with arity $q$. If the PCP has completeness $c$ and soundness $s$, then the language accepted by the PCP is reducible to a promise version of an arrity $q$ MaxCSP problem where in a 'no' instance at most an $s$ fraction of the constraints can be satisfied, and in a 'yes' instance at least a $c$ fraction of the constraints can be satisfied. These reductions are standard, look up any reference on PCPs.