  For a log scheme locally of finite type over $\mathbb{C}$, a natural
candidate for its profinite homotopy type is the profinite completion of its
Kato-Nakayama space. Alternatively, one may consider the profinite homotopy
type of the underlying topological stack of its infinite root stack. Finally,
for a log scheme not necessarily over $\mathbb{C}$, another natural candidate
is the profinite \'etale homotopy type of its infinite root stack. We prove
that, for a fine saturated log scheme locally of finite type over $\mathbb{C}$,
these three notions agree. In particular, we construct a comparison map from
the Kato-Nakayama space to the underlying topological stack of the infinite
root stack, and prove that it induces an equivalence on profinite completions.
In light of these results, we define the profinite homotopy type of a general
fine saturated log scheme as the profinite \'etale homotopy type of its
infinite root stack.
