This paper explores how singular sensitivity and logistic source terms affect finite-time blow-up in a parabolic-elliptic chemotaxis system. Specifically, we prove that when the logistic damping coefficient k is close to 1 and the singular sensitivity index satisfies a suitable condition, the corresponding radial solution blows up in finite time for large initial data or large sensitivity coefficient. This finding stands in contrast to the boundedness results established by Zhao: (i) classical solutions are bounded for all initial data when k>2 (2020, arXiv), and (ii) generalized solutions are bounded for suitably small initial data when k<2 (Czechoslovak Mathematical Journal, 2024). Moreover, our result extends Winkler' s work (2018, ZAMP) to the chemotaxis model with singular sensitivity; in particular, in the three-dimensional case, the range of k covered by our paper is wider than that in Winkler' s study.