Question 219·Hard·Nonlinear Functions

A function can be written in the form $f(x) = a b^x + c$, where $a$, $b$, and $c$ are real constants with $a \ne 0$ and $b > 1$. Suppose $f(1) - f(0) = 12$ and $f(3) - f(2) = 108$. What is the value of $b$?