2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139182It is known that a subharmonic function of finite order $ρ$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an approximation is made more precise, i. e. a constant $C$ decreases, then, beginning with $C=ρ/4$, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and functions subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov.12 pages, LATEXComplex Variables31A05, 30E10text