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# 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:

1. logb ac = logba + logbc
2. logb a/c = logba − logbc
3. logb ac = clogba
4. logb a = (logca)/logcb
5. blogc a = alogcb
6. (ba)c = bac
7. ba bc = ba+c
8. ba /bc = ba−c

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” ...

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