Title: Projective embedding of pairs and logarithmic K-stability,
Abstract: Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over
$D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed
complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler
form is in the cohomology class of a rational divisor $A$ and when $L$ has
negative CSCK metric on $D$, we show that the Kodaira embedding induced by
orthonormal basis of the Bergman space of $kA$ is almost balanced. As a
corollary, $(\hat{L},D,cA,0)$ is K-semistable.