  We investigate the existence and concentration of normalized solutions for a
$p$-Laplacian problem with logarithmic nonlinearity of type \[
  \left\{
  \begin{array}{ll}
  \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{p-2}u=\lambda
|u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline
  \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N,
  \end{array}
  \right. \] where $a,\varepsilon> 0$, $\lambda\in\mathbb R$ is known as the
Lagrange multiplier, $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla
\cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in
\mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable
assumptions. We prove that the number of positive solutions depends on the
profile of $V$ and each solution concentrates around its corresponding global
minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using
variational method.
  Moreover, we also get the existence of normalized solutions for some
logarithmic $p$-Laplacian equations involving mass-supercritical
nonlinearities.
