Data Structures and Algorithms in C++, Second Edition

Appendix A. Useful Mathematical Facts

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

Logarithms and Exponents

Logarithms and Exponents

The logarithm function is defined as

The following identities hold for logarithms and exponents:

1. log_bac = log_ba + log_bc

2. log_ba/c = log_ba − log_bc

3. log_ba^c = clog_ba

4. log_ba = (log_ca)/log_cb

5. 
6. 

7. b^ab^c = b^a+c

8. 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 < c²/4. We can write

The natural logarithm function lnx = log_ex, 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 A.2: If x ...