# Appendix B

# Useful Mathematical Facts

In this appendix we give several useful mathematical facts. We begin with some combinatorial definitions and facts.

#### Logarithms and Exponents

The logarithm function is defined as

The following identities hold for logarithms and exponents:

- log
_{b} *ac* = log_{b}a + log_{b}c
- log
_{b} *a*/*c* = log_{b}a − log_{b}c
- log
_{b} *a*^{c} = *c*log_{b}a
- log
^{b} *a* = (log_{c}a)/log_{c}b
*b*^{logc a} = *a*^{logcb}
- (
*b*^{a})^{c} = *b*^{ac}
*b*^{a} *b*^{c} = *b*^{a+c}
*b*^{a} /*b*^{c} = *b*^{a−c}

In addition, we have the following:

**Proposition B.1:** *If a* > 0, *b* > 0, *and c* > *a* + *b, then*

log*a* + log*b* < 2log*c* − 2.

**Justification:** It is enough to show that *ab* < *c*^{2}/4. We can write

The *natural logarithm* function ln*x* = log_{e} x, where *e* = 2.71828. . ., is the value of the following progression:

In addition,

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*,

#### Integer Functions and Relations

The “floor” ...