# NP characterization

I am asked to prove that the following are equivalent:

1. $$A \in NP$$
2. There exists $$B \in L= LOGSPACE$$ and $$c \geq 0$$ such that $$A = \{ x : \exists y \in \{0,1\}^* \text{ s.t. } |y| \leq |x|^c \text{ and } \langle x,y \rangle \in B \}$$

I know how to do it if instead of asking for logarithmic space, it asks for polynomical time. But I have no Idea how to do it in this case. In particular I am interested on the proff of 1) $$\Longrightarrow$$ 2) (the other is triviall because $$L \subseteq P$$.