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:

log

= log_{b}ac+ log_{b}a_{b}clog

/_{b}a*c*= log− log_{b}a_{b}clog

=_{b}a^{c}*c*log_{b}alog

= (log_{b}a)/log_{c}a_{c}b*b*=^{a}b^{c}*b*^{a+c}*b*/^{a}*b*=^{c}*b*^{a−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* < *c*^{2}/4. We can write

The natural logarithm function ln*x* = log* _{e}x*, where

In addition,

There are a number of useful inequalities relating to these functions (which derive from these definitions).

**Proposition A.2:** *If x ...*

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