  In the present paper, we consider elliptic operators
$L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $\Gamma$
with small enough constant, and whose coefficients $A$ satisfy a weak form of
the Dahlberg-Kenig-Pipher condition of approximation by constant coefficient
matrices, with a small enough Carleson norm, and show that the elliptic measure
with pole at infinity associated to $L$ is $A_\infty$-absolutely continuous
with respect to the surface measure on $\Gamma$, with a small $A_\infty$
constant. In other words, we show that for relatively flat uniformly
rectifiable sets and for operators with slowly oscillating coefficients the
elliptic measure satisfies the $A_\infty$ condition with a small constant and
the logarithm of the Poisson kernel has small oscillations.
