  In this paper, we study the parabolic-elliptic Keller-Segel system with
singular sensitivity and logistic-type source: $ u_t=\Delta
u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the
non-flux boundary conditions in a smooth bounded convex domain
$\Omega\subset\mathbb{R}^n$, $\chi,r,\mu>0$, $k>1$ and $n\ge 2$. It is shown
that the system possesses a globally bounded classical solution if
$k>\frac{3n-2}{n}$, and $r>\frac{\chi^2}{4}$ for $0<\chi\le 2$, or $r> \chi-1$
for $\chi>2$. In addition, under the same condition for $r,\chi$, the system
admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3n-2}{n}]$,
moreover this global generalized solution should be globally bounded provided
$\frac{r}{\mu}$ and the initial data $u_0$ suitably small.
