The partial decay rate $\Gamma(Z\to b\bar{b})$ is significantly
influenced by the mass of the top quark due to electroweak radiative
corrections. The leading $\sim m_t^2$ and the next-to-leading
contribution $\sim \ln m_t^2$ are known to be numerically of similar
size. In this work we calculate the QCD corrections to the
logarithmic correction using the heavy top mass expansion. The ${\cal
O} (\alpha\as \ln m_t^2)$ corrections are of the same order as the QCD
corrections to the quadratic top mass term, but of different sign.