The <formal additive group> and <formal multiplicative group> have laws
$$
\widehat{\mathbb G}_a(X,Y)=X+Y,
\qquad
\widehat{\mathbb G}_m(X,Y)=X+Y+XY.
$$
If $R$ is an <algebra over a field> whose scalar field is $\mathbb Q$, the series
$$
h(X)=\exp(X)-1
$$
has linear coefficient one and satisfies
$$
h(X+Y)=h(X)+h(Y)+h(X)h(Y).
$$
It is therefore the <exponential isomorphism between the additive and multiplicative formal groups>, with inverse $\log(1+X)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 5 / b / Solution

Over an <algebra over a field> whose scalar field is $\mathbb Q$, the series $\exp(X)-1$ defines an isomorphism from the <formal additive group> to the <formal multiplicative group>, with inverse $\log(1+X)$.

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 Incoming links: Exponential isomorphism between the additive and multiplicative formal groups