  We show that for $d\ge d_0(\epsilon)$, with high probability, the random
graph $G(n,d/n)$ contains an induced path of length
$(3/2-\epsilon)\frac{n}{d}\log d$. This improves a result obtained
independently by Luczak and Suen in the early 90s, and answers a question of
Fernandez de la Vega. Along the way, we generalize a recent result of Cooley,
Dragani\'c, Kang and Sudakov who studied the analogous problem for induced
matchings.
