  We propose and analyze two regularized finite difference methods for the
logarithmic Klein-Gordon equation (LogKGE). Due to the blowup phenomena caused
by the logarithmic nonlinearity of the LogKGE, it is difficult to construct
numerical schemes and establish their error bounds. In order to avoid
singularity, we present a regularized logarithmic Klein-Gordon equation
(RLogKGE) with a small regularized parameter $0<\varepsilon\ll1$. Besides, two
finite difference methods are adopted to solve the regularized logarithmic
Klein-Gordon equation (RLogKGE) and rigorous error bounds are estimated in
terms of the mesh size $h$, time step $\tau$, and the small regularized
parameter $\varepsilon$. Finally, numerical experiments are carried out to
verify our error estimates of the two numerical methods and the convergence
results from the LogKGE to the RLogKGE with the linear convergence order
$O(\varepsilon)$.
