Exponential Functions

Section 1.1 Problems

1.2 Elementary Functions

1.2.5 Exponential Functions

In our study of exponential functions, let’s first look at an example that illustrates where they occur.

EXAMPLE 9 Exponential Growth Bacteria reproduce asexually by cellular fission, in which the parent cell splits into two daughter cells after duplication of the genetic material. This division may happen as often as every 20 minutes; under ideal conditions, a bacterial colony can double in size in that time.

Let us measure time such that one unit of time corresponds to the doubling time of the colony. If we denote the size of the population at timet by N(t), then the function

N(t)=2^t, t ≥0 has the property of doubling its value every unit of time

N(t +¹)=^2t+1=²·^2t =²N(t) ^(1.2) The functionN(t)=^2t,t ≥0, is an exponential function because the variabletis in the exponent. We call the number 2 the base of the exponential functionN(t)=^2t. We find that when t = ^0, N(⁰) = 1; that is, there is just one individual in the population at timet = 0. If, at timet = 0, 40 individuals were present in the population, we would writeN(⁰)=^{40 and}

N(t)=40·2^t, t ≥0 (1.3)

You can verify thatN(t)in (1.3) also satisfiesN(t +¹)=²N(t)^.

It is often desirable not to specify the initial number of individuals in the equation describing N(t). This approach has the advantage that the equation for N(t)^then describes a more general situation, in the sense that we can use the same equation for different initial population sizes. We often denote the population size at time 0 byN₀(read “N sub 0”) instead ofN(0). The equation forN(t)is then

N(t)= N₀2^t, t ≥⁰

We can verify thatN(⁰) = N₀2⁰ = N₀ and that N(t +¹) = N₀2^t+1 = ²(N₀2^t) = 2N(t).

The function f(t)=2^tcan be defined for allt ∈R; its graph is shown in Figure 1.23.

1 0 2

3 1 2 3 4 5 2^t

t 5

10 15 20 25 30 35 f(t)

Figure 1.23 The function f(t)=2^t,t∈R.

Here is the definition of an exponential function:

Definition The function f is anexponentialfunction with baseaif f(x)=a^x

whereais a positive constant other than 1. The largest possible domain of f isR.

Whena = ^1, f(x) = 1 for all values of x. This is a case that will occur in biological examples, but is excluded from the definition since it is simply the constant function.

3 2 1 0 1 2 3 2

4 6 8

10y 2^x

x (1/3)^x

Figure 1.24 Exponential growth and exponential decay.

The basic shape of the exponential function f(x)= a^xdepends on the basea; two examples are shown in Figure 1.24. Asxincreases, the graph of f(x)= ^{2xshows a} rapid increase, whereas the graph of f(x)=(¹/³)^xshows a rapid decrease toward 0.

We find the rapid increase whenevera>1 and the rapid decrease whenever 0<a<

1. Therefore, we say that we have exponentialgrowthwhena > 1 and exponential decaywhen 0<a<^1.

Recall thata⁰ = 1 anda^1/k = √^k

a, wherekis a positive integer. In Subsection 1.1.5, we summarized the properties of exponentials. Since they are very important, we list them again here:

a^ra^s =a^r+s a^r

a^s =a^r−s a^−r = ¹

a^r a^rs

=a^{r s}

In many applications, the exponential function is expressed in terms of the base e = ².⁷¹⁸. . ., which we encountered in Subsection 1.1.5. The numbere is called thenatural exponential base. The exponential function with baseeis alternatively written as exp(x)^{. That is,}

exp(x)=e^x

The advantage of this alternative form can be seen when we try to write something likee^x2/

√x³+1: exp(x²/

x³+¹)is easier to read. More generally, ifg(x)is a function inx, then we can write, equivalently,

exp[g(x)] ^or e^g(x)

Bases 2 and 10 are also frequently used; in calculus, however,ewill turn out to be the most common base.

The next two examples provide an important application of exponential functions.

EXAMPLE 10 Radioactive Decay Radioactive isotopes such as carbon 14 are used to determine the absolute age of fossils or minerals, establishing an absolute chronology of the geological time scale. This technique was discovered in the early years of the 20th century and is based on the property of certain atoms to transform spontaneously by giving off protons, neutrons, or electrons. The phenomenon, calledradioactive

decay, occurs at a constant rate that is independent of environmental conditions. The method was used, for instance, to trace the successive emergence of the Hawaiian islands, from the oldest, Kauai, to the youngest, Hawaii (which is about 100,000 years old).

Carbon 14 is formed high in the atmosphere. It is radioactive and decays into nitrogen (N¹⁴). There is an equilibrium between atmospheric carbon 12 (C¹²) and carbon 14 (C¹⁴)—a ratio that has been relatively constant over a fairly long period.

When plants capture carbon dioxide (CO₂) molecules from the atmosphere and build them into a product (such as cellulose), the initial ratio of C¹⁴to C¹²is the same as that in the atmosphere. Once the plants die, however, their uptake of CO₂ceases, and the radioactive decay of C¹⁴causes the ratio of C¹⁴to C¹²to decline. Because the law of radioactive decay is known, the change in ratio provides an accurate measure of the time since the plants’ death.

According to the radioactive decay law, if the amount of C¹⁴at timetis denoted byW(t), withW(0)=W₀, then

W(t)=W0e^−λt, t ≥⁰

whereλ > ^{0 (}λis the lowercase Greek letter lambda) denotes thedecay rate. The functionW(t)=W0e^−λt is another example of an exponential function. Its graph is shown in Figure 1.25.

W(t) W(t) W₀e^lt W₀

W0

2

00 T_h t

Figure 1.25 The function W(t)=W₀e^−λt.

Frequently, the decay rate is expressed in terms of thehalf-lifeof the material, which is the length of time that it takes for half of the material to decay. If we denote this time byT_h, then (see Figure 1.25)

W(T_h)= ¹

2W₀=W₀e^−λTh from which we obtain

1

2 =e^−λTh 2=e^λTh

Recall from algebra (or Subsection 1.1.5) that, to solve for the exponent λT_h, we must take logarithms on both sides. Since the exponent has basee, we use natural logarithms and find that

ln 2=λTh

Solving forT_h orλ^yields

Th = ^{ln 2}

λ ^or λ= ^{ln 2}

Th

It is known that the half-life of C¹⁴is 5730 years. Hence,

λ= ^{ln 2}

5730 years

Note that the unit “years” appears in the denominator. It is important to carry the units along. When we computeλt in the exponent ofe^−λt, we need to measuret in units of years in order for the units to cancel properly. For example, supposet =2000 years; then

λt = ^{ln 2}

5730 years2000 years= (^{ln 2})(²⁰⁰⁰)

5730 ≈⁰.²⁴¹⁹

and we see that “years” appears in both the numerator and the denominator and thus can be canceled.

An application of the C¹⁴dating method is given in the next example.

EXAMPLE 11 Suppose that, on the basis of their C¹² content, samples of wood found in an archeological excavation site contain about 23% as much C¹⁴ as does living plant material. Determine when the wood was cut.

Solution

The ratio of the current amount of C¹⁴ to the amount of living plant material is expressed as

0.23= W(t) W(⁰) =e^−λt Taking logarithms (basee) on both sides, we obtain

ln(0.23)= −λt or

λt = −^ln(⁰.²³)=^{ln 1} 0.²³ Withλ=ln 2/(5730 years)from Example 10,

t = ^{5730 years} ln 2 ln 1

0.²³

Using a calculator to compute this result, we find that the wood was cut about 12,150 years ago.