Let $R$ be a topological ring of characteristic zero. Assume that $R$ is commutative. Let $G :R \times R \to R$ be a continuous function satisfying $G(G(x,y),z)=G(x,G(y,z))$ and $G(0,x)=x$, for all $x,y,z \in R$.

If we assume that $G$ is a convergent power series then $G$ is also a formal group law and the Honda's argument(cf. Hazewinkel, "Formal Groups and Applications", Chapter 1, 5.8, p.37) proves that G admits a logarithm, i.e. $G(x,y)=h^{-1}(h(x)+h(y))$.

**Question :** Does $G$ admit a logarithm if it is only continuous?

**EDIT** Assuming that $G$ is continuous, what are the additional assumptions that should be made on $R$ and $G$ to guarantee the existence of logarithms?