  We explain how logarithmic structures select principal components in an
intersection of schemes. These manifest in Chow homology and can be understood
using strict transforms under logarithmic blowups. Our motivation comes from
Gromov--Witten theory. The \textit{toric contact cycles} in the moduli space of
curves parameterize curves that admit a map to a fixed toric variety with
prescribed contact orders. We show that they are intersections of virtual
strict transforms of double ramification cycles in blowups of the moduli space
of curves. We supply a calculation scheme for the virtual strict transforms,
and deduce that toric contact cycles lie in the tautological ring of the moduli
space of curves. This is a higher dimensional analogue of a result of
Faber--Pandharipande. The operational Chow rings of Artin fans play a basic
role, and are shown to be isomorphic to rings of piecewise polynomials on
associated cone complexes. The ingredients in our analysis are Fulton's blowup
formula, Aluffi's formulas for Segre classes of monomial schemes, piecewise
polynomials, and degeneration methods. A model calculation in toric
intersection theory is treated without logarithmic methods and may be read
independently.
