  In this paper we investigate the inflationary phenomenology of an
Einstein-Gauss-Bonnet theory with the extension of a logarithmic modified
$f(R)$ gravity, compatible with the GW170817 event. The main idea of our work
is to study different results for an almost linear Ricci scalar through
logarithmic corrections and examine whether such model is viable. First of all,
the theoretical framework under slow-roll evolution of the scalar field is
presented and also developed the formalism of the constant-roll evolution
making predictions for the non- Gaussianities of the models is developed ,
since the constant-roll evolution is known to enhance non-Gaussianities. As
shown, the non-Gaussianities are of the order $\mathcal{O}\sim(10^{-1})$.
Furthermore, the slow-roll indices and the observational indices of inflation,
are calculated for several models of interest. As demonstrated, the
phenomenological viability of the models at hand is achieved for a wide range
of the free parameters and the logarithmic term has a minor contribution to
numerical calculations, as expected.
