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