In this paper, we develop a quantified propositional proof systems that corresponds to logarithmic-space reasoning. We begin by defining a class ΣCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF(2) formulas and no cut formula that is not quantifier free contains a free variable that does not appear in the final formula. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of V L into a family of tautologies that have polynomial-size GL ∗ proofs. V L is a theory of bounded arithmetic that is known to correspond to logarithmic-space reasoning. To do the translation, we find an appropriate axiomatization of V L, and put V L proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in V L. This is done by giving a logarithmic-space algorithm that witnesses GL ∗ proofs. 1