  We consider the existence and nonexistence of positive solution for the
following Br\'ezis-Nirenberg problem with logarithmic perturbation:
  \begin{equation*}
  \begin{cases}
  -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2} &x\in
\Omega,
  \quad \;\:\, u=0& x\in \partial \Omega,
  \end{cases}
  \end{equation*}
  where $\Omega$ $\subset$ $\R^N$ is a bounded smooth domain, $\lambda, \mu \in
\R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent
for the embedding $H^1_{0}(\Omega)\hookrightarrow L^{2^\ast}(\Omega)$. The
uncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest in
itself. We will show the existence of positive ground state solution which is
of mountain pass type provided $\lambda\in \R, \mu>0$ and $N\geq 4$. While the
case of $\mu<0$ is thornier. However, for $N=3,4$ $\lambda\in (-\infty,
\lambda_1(\Omega))$, we can also establish the existence of positive solution
under some further suitable assumptions. And a nonexistence result is also
obtained for $\mu<0$ and
$-\frac{(N-2)\mu}{2}+\frac{(N-2)\mu}{2}\log(-\frac{(N-2)\mu}{2})+\lambda-\lambda_1(\Omega)\geq
0$ if $N\geq 3$. Comparing with the results in Br\'ezis, H. and Nirenberg, L.
(Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the
parameter $\mu$ on logarithmic perturbation is not zero.
