  We study the nuclear structure function $F_2^A$ and its logarithmic
derivative in the high energy limit (small $x$ region) using the Color Glass
Condensate formalism. In this limit the structure function $F_2$ depends on the
quark anti-quark dipole-target scattering cross section $N_F (x_{bj}, r_t,
b_t)$. The same dipole cross section appears in single hadron and hadron-photon
production cross sections in the forward rapidity region in deuteron
(proton)-nucleus collisions at high energy, i.e. at RHIC and LHC. We use a
parameterization of the dipole cross section, which has successfully been used
to describe the deuteron-gold data at RHIC, to compute the nuclear structure
function $F_2^A$ and its log $Q^2$ derivative (which is related to gluon
distribution function in the double log limit). We provide a quantitative
estimate of the nuclear shadowing of $F_2^A$ and the gluon distribution
function in the kinematic region relevant to a future Electron-Ion Collider.
