In the present paper we consider elliptic operators
L
=
−
d
i
v
(
A
∇
)
L=-div(A\nabla )
in a domain bounded by a chord-arc surface
Γ
\Gamma
with small enough constant, and whose coefficients
A
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
L
is
A
∞
A_\infty
-absolutely continuous with respect to the surface measure on
Γ
\Gamma
, with a small
A
∞
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
∞
A_\infty
condition with a small constant and the logarithm of the Poisson kernel has small oscillations.