2.25 Use differentials to approximate: a) fi, h) ;/9997, c) (lO.OO3)i.

CHAPTER 5

Exponents and

Logarithms

In the last three chapters, we dealt exclusively with relationships expressed by

polynomial functions or by quotients of polynomial functions. However, in many

economics models, the function which naturally models the growth of a given

economic or financial variable over time has the independent variable t appearing

as an exponenl; for example, f(t) = 2™. These exponential functions occur nat-

urally, for example, as models for the amount of money in an interest-paying

savings account OT for the amount of debt in a fixed-rate mortgage account after f

Y.ZXS.

This chapter focuses on exponential functions and their derivatives. It also

describes the inverse of the exponential function-the logarithm, which can turn

multiplicative relationships between economicvariables into additive relationships

that are easier to work with. This chapter closes with applications of exponentials

and logarithms to problems of present value, annuities, and optimal holding time.

5.1 EXPONENTIAL FUNCTIONS

When first studying calculus, one works with a rather limited collection of func-

tional forms: polynomials and rational functions and their generalizations to frac-

tional and negative exponents-all functions constructed by applying the usual

arithmetic operations to the monomials ax™. We now enlarge the class of functions

under study by including those functions in which the variable x appears as an

exponent. These functions are naturally called exponential functions.

A simple example is f(x) = 2™, a function whose domain is all the real

numbers. Recall that:

(1) if x is a positive integer, 2™ means “multiply 2 by itself x times”;

(2) if x = I), 2” = 1, by definition;

(3) if x = l/n, 2”” = ;iz, the nth root of 2;

(4) if x = m/n, 2”“™ = ($@, the mth power of the nth root of 2; and

(5) if x is a negative number, 2^ means l/21x1, the reciprocal of 2™,“.

In these cases, the number 2 is called the base of the exponential function.

To understand this exponential function better, let™s draw its graph. Since we

do not know how to take the derivative of 2™ yet - (2”)™ is certainly not x2™-™ -

82

-3 l/X

-2 l/4

-1 l/2

Cl I

1 2

Table

2 4

5.1

3 8

we will have to plot points. We compute values of 2* in Table 5.1 and draw the

corresponding graph in Figure 5.1.

Note that the graph has the negativex-axis as a horizontal asymptote, but unlike

any rational function, the graph approaches this asymptote in only one direction.

In the other direction, the graph increases very steeply. In fact, it increases more

rapidly than any polynomial -“exponentially fast.”

In Figure 5.2, the graphs of fi(n) = 2*, fr(n) = 3”, and h(x) = 1tY arc

sketched. Note that the graphs are rather similar; the larger the base, the more

quickly the graph becomes asymptotic to the x-axis in one direction and steep in

the other direction.

The three bases in Figure 5.2 are greater than 1. The graph of y = b” is a bit

different if the base b lies between 0 and 1. Consider h(r) = (l/2)” as an example.

Table 5.2 presents a list of values of (x, y) in the graph of h for small integers n.

Note that the cntrics in the y-column of Table 5.2 arc the ˜amc as the cntrics in the

y-column of Table 5.1, but in reverse order, because (l/2)™ = 2-*. This means

that the graph of /l(r) = (l/2)™ is simply the reflection of the graph of f(x) = 2”

in the v-axis. as pictured in Figure 5.3. The graphs of (l/3)™ and (l/10)” look

similar to that of (I /?)˜I.

Figure

5.1

Figure

The graphs off,(x) = 2™, f&x) = 3™, and f&r) = 1Cf

5.2

-3 8

-2 4

-1 2

0 I

1 l/2

Table 2 l/4

5.2 3 I /8

t

\

Figure

Thr gruph of y = (l/2)

5.3

84

Negative bases are not allowed for the exponential function. For example, the

function k(x) = (-2)x would take on positive values for n an even integer and

negative values for x an odd integer; yet it is never zero in between. Furthermore,

since you cannot take the square root of a negative number, the function (-2)x is

not even defined for x = l/2 or, more generally, wheneverx is a fractionp/q and

y is an even integer. So, we can only work with exponential functions ax, where a

is a number greater than 0.

5.2 THE NUMBER e

Figure 5.2 prcscntcd graphs of exponential functions with bases 2, 3, and 10,

respectively. We now introduce a number which is the most important base for

an exponential function. the irrational number C. To motivate the definition of e,

consider the most basic economic situation-the growth of the investment in a

savings account. Suppose that at the beginning of the year, we deposit $A into a

savings account which pays interest at a simple annual interest late r. If WC will

let the account grow without deposits or withdrawals, after one year the account

will grow to A + rA = A( I + r) dollars. Similarly, the amount in the account in

any one year is (I + I) times the previous year™s amount. After two years, there

will be

A(1 + r)(l + 1.) = A(1 + r)™

d˜˜llars in the account. After I years, there will beA(1 + r)™ dollars in the account.

Next, suppose that the bank compounds interest four times a year; at the end of

each quarter. it pays interest at r-/4 times the current principal. After one quarter

of a year. the account contains A + iA dollars. After one year, that is, after four

compoundingsl there will be A (I + i)” dollars in the account. After 1 years, the

account will grow toA(l + 5)” dollars.

More generally. if interest is compounded N times a year, there will bcA( I + L)

dollars in the account after the first compounding period, A(1 + 5)” dollars in the

account after the first year, andA(1 + t)” dollars in the account after f years.

Many banks compound interest daily; others advertise that they compound

interest continuously. By what factor does money in the bank grow in one year at

interest rate r if interest is compounded so frequently. that is; if II is very large?

Mathematically, we are asking, “What is the limit of (I + i)” as n - E?” To

simplify this calculation, let™s begin with a 100 percent annual interest rate; that

is, I = 1. Some countries, like Israel, Argentina, and Russia, have expcricnccd