  We calculate the logarithmic corrections to the matrix elements for
$K^+\to\pi^+$ and $K\to$vacuum (which are used on the lattice to determine
$K\to\pi\pi$ amplitudes), in one-loop quenched and unquenched Chiral
Perturbation Theory. We find that these corrections can be large. We also
discuss, and present some results for, the direct determination of $K\to\pi\pi$
amplitudes. In particular, we address effects from choosing $m_s=m_d$ and
vanishing external spatial momenta, finite volume and quenching. In the
quenched octet case, we find enhanced finite-volume contributions which may
make numerical estimates of this matrix element unreliable for large volumes.
