  This paper is devoted to find the feasible shape functions for the
construction of static wormhole geometry in the frame work of
logarithmic-corrected $R^2$ gravity model. We discuss the asymptotically flat
wormhole solutions sustained by the matter sources with anisotropic pressure,
isotropic pressure and barotropic pressure. For anisotropic case, we consider
three shape functions and evaluate the null energy conditions and weak energy
conditions graphically along with their regions. Moreover, for barotropic and
isotropic pressures, we find shape function analytically and discuss its
properties. For the formation of traversable wormhole geometries, we cautiously
choose the values of parameters involved in $f(R)$ gravity model. We show
explicitly that our wormhole solutions violates the non-existence theorem even
with logarithmic corrections. We discuss all physical properties via graphical
analysis and it is concluded that the wormhole solutions with relativistic
formalism can be well justified with logarithmic corrections.
