  We write down a new "logarithmic" quasicoherent category
$\operatorname{Qcoh}_{log}(U, X, D)$ attached to a smooth open algebraic
variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This
is a (large) symmetric monoidal Abelian category, which we argue can be thought
of as the categorical substrate for logarithmic Hodge theory of $U$. We show
that its Hochschild homology theory coincides with the theory of log-forms on
$X$ with logarithmic structure induced by $D$, and in particular, that the
noncommutative Hodge-to de Rham sequence on $\operatorname{Qcoh}_{log}(U, X,
D)$ recovers known log Hodge structure on the de Rham cohomology of the open
variety $U$. As an application, we compute the Hochschild homology of the
category of coherent sheaves on the infinite root stack of Talpo and Vistoli in
the toroidal setting. We prove a derived invariance result for this theory:
namely, that strictly toroidal changes of compactification do not change the
derived category of $\operatorname{Qcoh}_{log}(U, X, D)$. The definition is
motivated by the coherent object appearing in the author's microlocal mirror
symmetry result [20].
  In this paper, the first in a series, we work over an algebraically closed
field of characteristic zero. The next installment will develop the
characteristic p and mixed-characteristic theories.
