The Nature Of Exponential Functions

As introduced in connection with polynomial functions, the term exponent means an indicator of the power to which a variable is to be raised. In power expressions such as .v3 or x5, the exponents are constants; but there is no reason why we cannot also have a variable exponent, such as in 3X or 3', where the number 3 is to be raised to varying powers (various values of x). A function whose independent variable appears in the role of an exponent is called an exponential function.

Simple Exponential Function

In its simple version, the exponential function may be represented in the form (10.1) y=f(t) = bt (b> 1)

where y and t are the dependent and independent variables, respectively, and b denotes a fixed base of the exponent. The domain of such a function is the set of all real numbers. Thus, unlike the exponents in a polynomial function, the variable exponent t in (10.1) is not limited to positive integers—unless we wish to impose such a restriction.

But why the restriction of b > 1? The explanation is as follows. In view of the fact that the domain of the function in (10.1) consists of the set of all real numbers, it is possible for / to take a value such as j. If ¿> is allowed to be negative, the half power of b will involve taking the square root of a negative number. While this is not an impossible task, we would certainly prefer to take the easy way out by restricting b to be positive. Once we adopt the restriction b > 0, however, we might as well go all the way to the restriction b > 1: The restriction b > 1 differs from b > 0 only in the further exclusion of the cases of (1)0 < b < 1 and (2) b = 1; but as will be shown, the first case can be subsumed under the restriction b > 1, whereas the second case can be dismissed outright. Consider the first case. If b = j, then we have

This shows that a function with a fractional base can easily be rewritten into one with a base greater than 1. As for the second case, the fact that b = 1 will give us the function y — 1' = 1, so that the exponential function actually degenerates into a constant function; it may therefore be disqualified as a member of the exponential family.

Graphical Form

The graph of the exponential function in (10.1) takes the general shape of the curve in Fig. 10.1. The curve drawn is based on the value b = 2; but even for other values of /?, the same general configuration will prevail.

Several salient features of this type of exponential curve may be noted. First, it is continuous and smooth everywhere; thus the function should be everywhere differentiable. As a matter of fact, it is continuously differentiable any number of times. Second, it is monotonically increasing, and in fact y increases at an increasing rate throughout. Consequently, both the first and second derivatives of the function >> = bl should be positive—a fact we should be able to confirm after we have developed the relevant differentiation formulas. Third, we note that, even though the domain of the function contains negative as well as positive numbers, the range of the function is limited to the open interval (0, oo). That is, the dependent variable y is invariably positive, regardless of the sign of the independent variable t.

-4 -3 -2 -1 0 1 2 3 4 Figure 10.1

The monotonicity of the exponential function entails at least two interesting and significant implications. First, we may infer that the exponential function must have an inverse function, which is itself monotonic. This inverse function, we shall find, turns out to be a logarithmic function. Second, since monotonicity means that there is a unique value of / for a given value of y and since the range of the exponential function is the interval (0, oc), it follows that we should be able to express any positive number as a unique power of a base b > 1. This can be seen from Fig. 10.1, where the curve of y = 2! covers all the positive values of y in its range; therefore any positive value of y must be expressible as some unique power of the number 2. Actually, even if the base is changed to some other real number greater than 1, the same range holds, so that it is possible to express any positive number y as a power of any base b > 1.

Generalized Exponential Function

This last point deserves closer scrutiny. If a positive v can indeed be expressed as powers of various alternative bases, then there must exist a general procedure of base conversion. In the case of the function y = 9\ for instance, we can readily transform it into y ~ (32)' = 32r, thereby converting the base from 9 to 3. provided the exponent is duly altered from t to It. This change in exponent, necessitated by the base conversion, does not create any new type of function, for, if we let w = 2/, then y = 3l! = 3M is still in the form of (10.1). From the point of view of the base 3, however, the exponent is now 2t rather than t. What is the effect of adding a numerical coefficient (here. 2) to the exponent t!

The answer is to be found in Fig. 10.2a, where two curves are drawn - one for the functiony = f(t) = bl and one for another function y = g(t) = blt. Since the exponent in the latter is exactly twice that of the former, and since the identical base is adopted for the two functions, the assignment of an arbitrary value t = t(i in the function g and t ----- 2/0 in the function / must yield the same value:

Thus the distancey()J will be half of y{)K. By similar reasoning, for any value of y. the function g should be exactly halfway between the function / and the vertical axis. It may be concluded, therefore, that the doubling of the exponent has the effect of compressing the exponential curve halfway toward the y axis, whereas halving the exponent will extend the curve away from the y axis to twice the horizontal distance.

It is of interest that both functions share the same vertical intercept

The change of the exponent t to It. or to any other multiple of t. will leave the vertical intercept unaffected. In terms of compressing, this is because compressing a zero horizontal distance will still yield a zero distance.

The change of exponent is one way of modifying—and generalizing—the exponential function of (10.1); another is to attach a coefficient to b\ such as 2bl. [Warning: 2b! =h (2b)1.] The effect of such a coefficient is also to compress or extend the curve, except that this time the direction is vertical. In Fig. 10.2b, the higher curve represents^ = 2b\ and the lower one is y = b*. For every value of the former must obviously be twice as high, because it has a y value twice as large as the latter. Thus we have t0Jf — J'Kf. Note that the vertical intercept, too, is changed in the present case. We may conclude that doubling the coefficient (here, from 1 to 2) serves to extend the curve away from the horizontal axis to twice the vertical distance, whereas halving the coefficient will compress the curve halfway toward the t axis.

With the knowledge of the two modifications discussed above, the exponential function y = br can now be generalized to the form

(10.2) = abct where a and c are "compressing" or "extending" agents. When assigned various values, they will alter the position of the exponential curve, thus generating a whole family of exponential curves (functions). If a and c are positive, the general configuration shown in Fig. 10.2 will prevail; if a or c or both are negative, however, then fundamental modifications will occur in the configuration of the curve (see Exercise 10.1-5 below).

A Preferred Base

What prompted the discussion of the change of exponent from t to ct was the question of base conversion. But, granting the feasibility of base conversion, why y y

would one want to do it anyhow? One answer is that some bases are more convenient than others as far as mathematical manipulations are concerned.

Curiously enough, in calculus, the preferred base happens to be a certain irrational number denoted by the symbol e:

When this base e is used in an exponential function, it is referred to as a natural exponential function, examples of which are y = e' y = e3t y = AerI

These illustrative functions can also be expressed by the alternative notations y = exp(r) y — exp(3/) y = A exp(rt)

where the abbreviation exp (for exponential) indicates that e is to have as its exponent the expression in parentheses.

The choice of such an outlandish number as e = 2.71828... as the preferred base will no doubt seem bewildering. But there is an excellent reason for this choice, for the function el possesses the remarkable property of being its own derivative! That is, d_ dt el = e a fact which will reduce the work of differentiation to practically no work at all. Moreover, armed with this differentiation rule—to be proved later in this chapter —it will also be easy to find the derivative of a more complicated natural exponential function such as y = Aert. To do this, first let w = rt, so that the function becomes y = AeH where w = rt, and A, r are constants

Then, by the chain rule, we can write dt dw dt f

That is,

The mathematical convenience of the base e should thus be amply clear.

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