  We study the existence of normalized solutions to the following logarithmic
Schr\"{o}dinger equation \begin{equation*}\label{eqs01} -\Delta u+\lambda
u=\alpha u\log u^2+\mu|u|^{p-2}u, \ \ x\in\R^N, \end{equation*} under the mass
constraint \[ \int_{\R^N}u^2\mathrm{d}x=c^2, \] where $\alpha,\mu\in \R$, $N\ge
2$, $p>2$, $c>0$ is a constant, and $\lambda\!\in\!\R$ appears as Lagrange
multiplier. Under different assumptions on $\alpha,\mu,p$ and $c$, we prove the
existence of ground state solution and excited state solution. The asymptotic
behavior of the ground state solution as $\mu\to 0$ is also investigated. Our
results including the case $\alpha<0$ or $\mu<0$, which is less studied in the
literature.
