A sequence of positive real numbers a_1, a_2, ..., a_n, is log-concave if a_i^2 >= a_{i-1}a_{i+1} for all i ranging from 2 to n-1. Log-concavity naturally arises in various aspects of mathematics, each characterized by different underlying mechanisms. Examples range from inequalities that are readily provable, such as the binomial coefficients a_i = \binom{n}{i}, to intricate inequalities that have taken decades to resolve, such as the number of forests a_i in a graph G with i edges. It is then natural to ask if it can be shown that the latter type of inequalities is intrinsically more challenging than the former. In this talk, we provide a rigorous framework to answer this type of questions, by employing a combination of combinatorics, complexity theory, and geometry. This is a joint work with Igor Pak.