Theorem 2

Consider the case a > 1. Suppose x₁ and x₂ are finite and x₁ ≈ x₂. Say x₁ < x₂. Choose hyperrational numbers r₁ and r₂ infinitely close to x₁ and x₂ such that

r₁ < x₁ < x₂ < r₂.

The inequalities for exponents hold for hyperreal x by the Transfer Principle, so

a^r1 < a^x1 < a^X2 < a^r2

But r₁ ≈ r₂, so a^r1 ≈ a^r2. Therefore a^X1 ≈ a^X2, and y = a^x is continuous.

The case a ≤ 1 is similar.

An example of an exponential function is given by the growth of a population f(t) with a constant birth and death rate. It grows in such a way that the rate of change of the population is proportional to the population. Given an integer n, the population increase from time t to time t + 1/n is a constant times f(t).

f(t + 1/n) - f(t) = cf(t).

Then

f(t + 1/n) = kf(t)

where k = c + 1.

Let us set f(0) = 1; that is, we choose f(0) for our unit of population. Then

f(0) = 1, f(1/n) = k, f(2/n) = k² ..., f(m/n) = k^m. So if we put f(1) = a = kⁿ,

we have

f(m/n) = a^m/n.

We conclude that for any rational number m/n, the population at time t = m/n is a^m/n. In reality, of course, the population is not a continuous function of time because its value is always a whole number. However, it is convenient to approximate the population by the exponential function a^x, and to make a^x continuous by defining it for all real x.

If the birth rate of a population is greater than the death rate, the growth curve will be a^x where a > 1 and the population will increase. Similarly, if the birth and death rates are equal, a = 1 and the population is constant. When the death rate exceeds the birth rate, a < 1 and the population decreases.

Warning: A population grows exponentially only when the birth rate minus the death rate is constant. This rarely happens for long periods of time, because a large change in the population will itself cause the birth or death rate to change. For example, if the population of the earth quadrupled every century it would reach the impossible figure of one quadrillion, or 10¹⁵, people in about 900 years. In the 20th century the birth rate of the United States has fluctuated wildly while the death rate has decreased. Later in this chapter we shall discuss more realistic growth functions which grow nearly exponentially at first but then level off at a limiting value.

The inequalities for exponents can be used to get approximate values for a^b and to evaluate limits.