Let \(\varphi\) be a plurisubharmonic function on a complex manifold \(X\). Following A. M. Nadel [Ann. Math. (2) 132, No. 3, 549–596 (1990; Zbl 0731.53063)], one can define the multiplier ideal sheaf \({\mathcal I}(\varphi)\) (with weight \(\varphi\)) to be the sheaf of germs of holomorphic functions \(f\) such that \(|f|e^{-2\varphi}\) is locally integrable. The aim of this paper is to prove the following: Let \(X\) be a complex manifold (compact or noncompact) with dimension bigger than one and let \(z_0\in X\). Then there exists a quasi-plurisubharmonic function \(\varphi\) on \(X\) such that for any plurisubharmonic function \(\psi\geq\varphi\) near \(z_0\) with logarithmic poles, \[ c_{z_0}(\varphi) <c_{z_0}(\psi) \] holds, where \(c_{z_0}(\varphi) := \sup\left\{c|{\mathcal I}(c\varphi)_{z_0}={\mathcal O}_{z_0} \right\}\) is the complex singularity exponent of \(\varphi\). As a direct consequence, the author proves the nonexistence of decreasing equisingular approximations with logarithmic poles.