Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 3] FUNCTIONS AND ALGORITHMS 49

Arithmetic moduloMrefers to the arithmetic operations of addition, multiplication, and subtraction where
the arithmetic value is replaced by its equivalent value in the set

{ 0 , 1 , 2 ,...,M− 1 } or in the set { 1 , 2 , 3 ,...,M}

For example, in arithmetic modulo 12, sometimes called “clock” arithmetic,

6 + 9 ≡ 3 , 7 × 5 ≡ 11 , 1 − 5 ≡ 8 , 2 + 10 ≡ 0 ≡ 12

(The use of 0 orMdepends on the application.)

Exponential Functions

Recall the following definitions for integer exponents (wheremis a positive integer):

am=a·a···a(mtimes), a^0 = 1 ,a−m=

1

am

Exponents are extended to include all rational numbers by defining, for any rational numberm/n,

am/n=n

√

am=(n

√

a)m

For example,

24 = 16 , 2 −^4 =

1

24

=

1

16

, 1252 /^3 = 52 = 25

In fact, exponents are extended to include all real numbers by defining, for any real numberx,

ax=lim

r→x

ar, whereris a rational number

Accordingly, the exponential functionf(x)=axis defined for all real numbers.

Logarithmic Functions

Logarithms are related to exponents as follows. Letbbe a positive number. The logarithm of any positive
numberxto be the baseb, written
logbx

represents the exponent to whichbmust be raised to obtainx. That is,

y=logbx and by=x

are equivalent statements. Accordingly,

log 28 =3 since 2^3 = 8 ; log 10100 = 2 since 10^2 = 100

log 264 =6 since 2^6 = 64 ; log 100. 001 =−3 since 10−^3 = 0. 001

Furthermore, for any baseb, we haveb^0 =1 andb^1 =b; hence

logb 1 =0 and logbb= 1

The logarithm of a negative number and the logarithm of 0 are not defined.
Frequently, logarithms are expressed using approximate values. For example, using tables or calculators, one
obtains
log 10300 = 2 .4771 and loge 40 = 3. 6889

as approximate answers. (Heree= 2. 718281 ....)