This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, {ut=Δu−χ1∇⋅(uw∇w)+μ1u−μ1uα,x∈Ω,t>0,υt=Δv−χ2∇⋅(υw∇w)+μ2υ−μ2υβ,x∈Ω,t>0,wt=Δw−(u+υ)w,x∈Ω,t>0, $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi _{1}\nabla \cdot (\frac{u}{w}\nabla w)+\mu _{1}u-\mu _{1}u^{\alpha }, &x\in \varOmega , t>0, \\ \upsilon _{t}=\Delta v-\chi _{2}\nabla \cdot ( \frac{\upsilon }{w}\nabla w)+\mu _{2}\upsilon -\mu _{2}\upsilon ^{\beta } , &x\in \varOmega , t>0, \\ w_{t}=\Delta w-(u+\upsilon )w, &x\in \varOmega , t>0, \end{cases}\displaystyle \end{aligned}$$ under the homogeneous Neumann boundary conditions and for widely arbitrary positive initial data in a bounded domain Ω⊂Rn(n≥1) $\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)$ with smooth boundary, where χi $\chi _{i}$, μi>0 (i=1,2) $(i=1, 2)$ and α, β>1 $\beta >1$. It is proved that there exists a global classical solution if max{χ1,χ2}<2n,min{μ1,μ2}>n−2n,α=β=2 for n≥2 or any χi>0 (i=1,2) $(i=1,2)$, μi>0 (i=1,2) $(i=1,2)$, α, β>1 for n=1 $n=1$.