In this appendix we give several useful mathematical facts. We begin with some combinatorial definitions and facts.
The logarithm function is defined as
The following identities hold for logarithms and exponents:
In addition, we have the following:
Proposition B.1: If a > 0, b > 0, and c > a + b, then
loga + logb < 2logc − 2.
Justification: It is enough to show that ab < c2/4. We can write
The natural logarithm function lnx = loge x, where e = 2.71828. . ., is the value of the following progression:
There are a number of useful inequalities relating to these functions (which derive from these definitions).
Proposition B.2: If x > −1,
Proposition B.3: For 0 ≤ x < 1,
Proposition B.4: For any two positive real numbers x and n,
The “floor” ...