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:
logbac = logba + logbc
logba/c = logba − logbc
logbac = clogba
logba = (logca)/logcb
babc = ba+c
ba/bc = ba−c
In addition, we have the following.
Proposition A.1: If a > 0, b > 0, and c > a + b, then
Justification: It is enough to show that ab < c2/4. We can write
The natural logarithm function lnx = logex, 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 A.2: If x ...