Answer to Question #18662 in Abstract Algebra for john.george.milnor

Question

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). For any y ∈ G and α ∈ k, define y^α = exp(α log(y)) ∈ G. Show that log(yα) = α log(y), (y^α)^β = y^αβ, and y^α*y^β = y^(α+β) for any α, β ∈ k.