Reducing a P-complete search problem to an NP-complete decision problem

Let us say that a search problem $R$ has an A-reduction if there is a polynomial algorithm $A$ that solves $R$ using an oracle to the decision problem $S_R$, so that for each input $x$, the number of oracle calls that $A$ makes is at most $100log|x|$.

Assume that $P ne NP$. Prove or disprove:
There is a search problem $R in FNP$ so that $S_R$ is $NPC$ and $R$ has an A-reduction.

I am a bit confused about what “at most $100log|x|$” means. I also don’t know how to find a search problem $R$ that satisfies the requirement.