  A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse}
if there is an absolute constant $C$ so that for any positive integer $x$ the
sequence contains at most $C$ elements in the interval $[x,2x)$. In this note
we study arithmetic progressions in sums of log-sparse subsets of
$\mathbb{Z}_{> 0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$
of $\mathbb{Z}_{> 0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an
arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that
this is nearly tight by proving that there exist log-sparse sets $S_1, \dots,
S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size
$n^{(1-o(1)) n}.$
