  In this paper we establish the existence and multiplicity of nontrivial
solutions to the following problem \begin{align*} \begin{split}
(-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R},
\end{split} \end{align*} where $\mu>0$, $(*)$ is the convolution operation
between two functions, $0<\gamma<1$, $f$ is a function with a certain type of
growth. We prove the existence of a nontrivial solution at a certain mountain
pass level and another ground state solution when the nonlinearity $f$ is of
exponential critical growth.
