SE-SKOLORNAS 1978 Problem 4

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$ . (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.