  We prove a lemma, which we call the Order Ideal Lemma, that can be used to
demonstrate a wide array of log-concavity and log-convexity results in a
combinatorial manner using order ideals in distributive lattices. We use the
Order Ideal Lemma to prove log-concavity and log-convexity of various sequences
involving lattice paths (Catalan, Motzkin and large Schr\"oder numbers),
intervals in Young's lattice, order polynomials, specializations of Schur and
Schur Q-functions, Lucas sequences, descent and peak polynomials of
permutations, pattern avoidance, set partitions, and noncrossing partitions. We
end with a section with conjectures and outlining future directions.
