Nonassociative exponential and logarithm

We consider the unique power series E(x)=ex=exp(x) and L(x)=log(1+x) with rational coefficients in a nonassociative, noncommutative variable x defined with the properties E′(0)=1, E(x)•E(x)=E(2x), E′(x)=E(x) and L(0)=0, L′(0)=1, L(2x+x2)=2•L(x), where E′(x) and L′(x) are the formal derivatives of E(x) and L(x) with respect to x, respectively. These functions satisfy the relations log(ex)=x and exp(log(1+x))=1+x. In this paper we discuss elementary properties of exp and log. The set of nonassociative, noncommutative monomials in x is indexed by planar binary trees. Our main results provide formulas for the coefficients of exp and log derived by methods of combinatorics of trees.