  In the paper we consider the asymptotics of logarithmic tails of a perpetuity
$$R \stackrel{d}{=}\sum_{j=1}^\infty Q_j
\prod_{k=1}^{j-1}M_k,\qquad(M_n,Q_n)_{n=1}^\infty \mbox{ are i.i.d. copies of
}(M,Q),$$ in the case when $\mathbb{P}(M\in[0,1))=1$ and $Q$ has all
exponential moments. If $M$ and $Q$ are independent, under regular variation
assumptions, we find the precise asymptotics of $-\log\mathbb{P}(R>x)$ as
$x\to\infty$. Moreover, we deal with the case of dependent $M$ and $Q$ and give
asymptotic bounds for $-\log\mathbb{P}(R>x)$. It turns out that dependence
structure between $M$ and $Q$ has a significant impact on the asymptotic rate
of logarithmic tails of $R$. Such phenomenon is not observed in the case of
heavy-tailed perpetuities.
