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Let f be a transcendental entire function of finite lower order, and let $q_{\nu }$ be rational functions. For 0 < γ < ∞ let B(γ):= πγ/sinπγ if γ ≤ 0.5, B(γ):= πγ if γ > 0.5. We estimate the upper and lower logarithmic density of the set $r:{\sum }_{1\le \nu \le k}log⁺ma{x}_{\left|\right|z\left|\right|=r}{\left|f\left(z\right)-q_{\nu }\left(z\right)\right|}^{-1}<B\left(\gamma \right)T(r,f)$.