In this paper we consider the asymptotics of logarithmic tails of a perpetuity R=D∑j=1∞Qj∏k=1j-1Mk, where (Mn,Qn)n=1∞ are independent and identically distributed copies of (M,Q), for the case when ℙ(M∈[0,1))=1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of -logℙ(R>x) as x→∞. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for -logℙ(R>x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.
