  We provide scaling limits for the block counting process and the fixation
line of $\Lambda$-coalescents as the initial state $n$ tends to infinity under
the assumption that the measure $\Lambda$ on $[0,1]$ satisfies
$\int_{[0,1]}u^{-1}(\Lambda-b\lambda)({\rm d}u)<\infty$ for some $b>0$. Here
$\lambda$ denotes the Lebesgue measure. The main result states that the block
counting process, properly logarithmically scaled, converges in the Skorohod
space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The
result is applied to beta coalescents with parameters $1$ and $b>0$. We split
the generators into two parts by additively decomposing Lambda and then prove
the uniform convergence of both parts separately.
