  This is the first in a pair of papers developing a framework for the
application of logarithmic structures in the study of singular curves of genus
$1$. We construct a smooth and proper moduli space dominating the main
component of Kontsevich's space of stable genus $1$ maps to projective space. A
variation on this theme furnishes a modular interpretation for Vakil and
Zinger's famous desingularization of the Kontsevich space of maps in genus $1$.
Our methods also lead to smooth and proper moduli spaces of pointed genus $1$
quasimaps to projective space. Finally, we present an application to the log
minimal model program for $\mathcal{M}_{1,n}$. We construct explicit
factorizations of the rational maps among Smyth's modular compactifications of
pointed elliptic curves.
