  In the present work we obtain the existence and multiplicity of nontrivial
solutions for the Choquard logarithmic equation $(-\Delta)_{p}^{s}u +
|u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ }
\mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ \lambda
>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with
exponential critical and subcritical growth. We guarantee the existence of a
nontrivial solution at the mountain pass level and a nontrivial ground state
solution under critical and subcritical growth. Morever, when $ f $ has
subcritical growth we prove the existence of infinitely many solutions, via
genus theory.
