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

50 FUNCTIONS AND ALGORITHMS [CHAP. 3

Three classes of logarithms are of special importance: logarithms to base 10, calledcommon logarithms;
logarithms to basee, callednatural logarithms;and logarithms to base 2, calledbinary logarithms. Some texts
write
lnxfor logex and lgxor logxfor log 2 x

The term logx, by itself, usually means log 10 x; but it is also used for logexin advanced mathematical texts and
for log 2 xin computer science texts.
Frequently, we will require only the floor or the ceiling of a binary logarithm. This can be obtained by looking
at the powers of 2. For example,
⌊
log 2100

⌋

=6 since 2^6 =64 and 2^7 = 128

⌈

log 21000

⌉

=9 since 2^8 =512 and 2^9 = 1024

and so on.

Relationship between the Exponential and Logarithmic Functions

The basic relationship between the exponential and the logarithmic functions

f(x)=bx and g(x)=logbx

is that they are inverses of each other; hence the graphs of these functions are related geometrically. This relation-
ship is illustrated in Fig. 3-5 where the graphs of the exponential functionf(x)= 2 x, the logarithmic function
g(x)=log 2 x, and the linear functionh(x)=xappear on the same coordinate axis. Sincef(x)= 2 xand
g(x)=log 2 xare inverse functions, they are symmetric with respect to the linear functionh(x)=xor, in other
words, the liney=x.

Fig. 3-5

Figure3-5alsoindicatesanotherimportantpropertyoftheexponentialandlogarithmicfunctions.Specifically,
for any positivec, we have

g(c) < h(c) < f (c), that is, g(c)<c<f(c)

In fact, ascincreases in value, the vertical distancesh(c)−g(c)andf(c)−g(c)increase in value. Moreover,
the logarithmic functiong(x)grows very slowly compared with the linear functionh(x), and the exponential
functionf(x)grows very quickly compared withh(x).

3.5Sequences, Indexed Classes of Sets

Sequences and indexed classes of sets are special types of functions with their own notation. We discuss
these objects in this section. We also discuss the summation notation here.