  We provide bounds for the sequence of eigenvalues $\{\lambda_i(\Omega)\}_i$
of the Dirichlet problem $$ L_\Delta u=\lambda u\ \ {\rm in}\ \,
\Omega,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus \Omega,$$ where
$L_\Delta$ is the logarithmic Laplacian operator with Fourier transform symbol
$2\ln |\zeta|$. The logarithmic Laplacian operator is not positively definitive
if the volume of the domain is large enough. In this article, we obtain the
upper and lower bounds for the sum of the first $k$ eigenvalues by extending
the Li-Yau method and Kr\"oger's method respectively. Moreover, we show the
limit of the sum of the first $k$ eigenvalues, which is independent of the
volume of the domain. Finally, we discuss the lower and upper bounds of the
$k$-th principle eigenvalue, the asymptotic behavior of the limit of
eigenvalues.
