стр. 14 |

We compute (1 + A)вЂќ with a calculator for various values of n and list the

results in Table 5.3.

2.0

2.25

2.4414

2.59374

2.704814

2.716923Y

2.7181459

2.71826824

2.7lRZRlhY3

One sees in Table 5.3 that the sequence (I + hiвЂќ is an increasing sequence

in II and converges to a numhcr a littlc higgcr than 2.7. The limit turns out to he

an irrational numhcr, in that it cannot he witten as a fraction or as a repeating

decimal. The letter e is reserved to denote this number: Cormally.

To seven decimal places. e = 2.71X2X18вЂ™ ..

This number e plays the same fundamental role in tinance and in economics

that the number TT plays in geometry. In particular. the function f(.r) = rвЂ™ is called

I/X, exponential function and is frequently written as cxp(.v). Sincc 2 < o c 3.

the graph ofexp(.r) = c˜вЂ™ is shaped like the graphs in Figure 5.2.

Next. we reconsider the gcncral intcrcst rater and ask: What is the limit of the

seqвЂќencc

in tams of e? A simple change of variables answers this question. Fix IвЂ™ > I) for

the rest of this discussion. Let 111 = n/r: so II = mr. As II gets larger and goes to

infinity, so dots nz. (Rcmcmhcr is tixcd.) Since r/n = I /III,

IвЂ™

(l i ;)вЂќ = (I + ;)IвЂќ, = ((I + k)вЂњвЂ˜)вЂ™

[5.2] e 87

THE NUMBER

by straightforward substitution. Letting n - ˜0, we find

In the second step, we used the fact that nвЂ™ is a continuous function of x, so that

if {x,,,}˜=, is a sequence of numbers which converges to x0. then the sequence of

powers {XL} converges to x;; that is

(1

limx, вЂ™ = ;m, (XL)

m-z

If we let the account grow for f years, then

˜˜(l+;˜=˜˜((l+˜)II)

= (/&(I + ;)вЂњ)

The following theorem summarizes these simple limit computations

Theorem 5.1 As n - x, the sequence 1 + ,i converges to a limit denoted

( вЂ˜)вЂќ

by the symbol e. Furthermore,

вЂњ-z ( + k ) n = eвЂ™.

lim 1

n

If one deposits A dollars in an account which pays annual interest at rate r

compounded continuously, then after f years the account will grow to AeвЂќ

dollars.

Note the advantages of frequent compounding. At I = 1, that is, at a 100

percent interest rate, A dollars will double to 2.4 dollars in a year with no com-

pounding. However, if interest is compounded continuously, then the A dollars

will grow to eA dollars with e > 2.7; the account nearly triples in size.

88 EXPONENTS AND LOGARITHMS 151

5.3 LOGARITHMS

Consider a general exponential function, y = al, with hex u > I. Such an

exponential function is a strictly increasing function:

In words, the more times you multiply rr hy itself. the higgcr it gets. As we pointed

out in Theorem 4.1, strictly increasing functions have natural invcrscs. Recall that

the inverse of the function J = f(r) is the function obtained hy solving y = f(x)

for x in terms of 4вЂ˜. For cxamplc, for a > 0, the inverse of the increasing linear

function f(x) = a x + h i s t h e l i n e a r f u n c t i o n &) = (l/o)& h), w h i c h i s

computed by solving the equation y = ux + h for n in terms of y:

(2)

In a sense, the inverse g off undoes the operation off, so that

df(Q) = 1.

See Section 4.2 for a detailed discussion of the inverse of a function.

f(˜r) =

We cannot compute the inverse of the increasing cxponcntial function

(IвЂ™˜ explicitly because we canвЂ™t solve J = ˜1˜вЂ™ for in terms of ?. as did in (7).

I WC

However, this inverse function is important enough that give it a name. We call

WC

base II logarithm and write

it the

logarithm ofz; hy detinition, is the power to which one must raise u to yield

The

z. It follows immediat& from this definition that

&(Zl =z

and log,, (aвЂ™) = z. (3)

W C often write log,,(z) without parentheses_ as log,,:

Base 10 Logarithms

LetвЂ˜s tirst work with base u = IO. The logarithmic function for hasr IO is such

a commonly used logarithm that it is usually written as )вЂ™ = Log.x with an

uppercase L:

˜xxumplr5.1 Forexamplc, the Log of 1000 is that power of lOwhich yields 1000.

стр. 14 |