  We study the existence and concentration behavior of the bound states for the
following logarithmic Schr\"odinger equation \begin{equation*} \begin{cases}
-\varepsilon^2\Delta v+V(x)v=v\log v^2 \ \ &\text {in}\ \ \mathbb R^N,\\
v(x)\to 0 \ \ &\text {as}\ \ |x|\to\infty, \end{cases} \end{equation*} where
$N\geq 1$, $\varepsilon>0$ is a small parameter, and $V$ may be unbounded below
at infinity with a speed of at most quadratic strength. We show that around
various types of local topological critical points of the potential function,
positive bound state solutions exist and concentrate as $\varepsilon\to0$.
