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It is known that a subharmonic function of finite order $\rho$ can beapproximated 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 thatif such an approximation is made more precise, i. e. a constant  $C$ decreases, then, beginning with $C=\rho/4$, the size of the exceptionalset enlarges substantially. Similar results are proved for subharmonicfunctions of infinite order and functions subharmonic in the unit disk. Thesetheorems improve and complement a result by Yulmukhametov.

Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic ones in Chebyshev metrics