Exponential Equations II: _{The constant e and limits to growth}

The typical form of exponential equations in science

Many natural phenomena exhibit exponential growth (like population increase) or decay (like the depletion of radioactive isotopes), and thus exponential equations are frequently used in science. In virtually all cases, time is an important variable in these phenomena, so scientists often use a particular exponential equation with the variable of time already built into it. The exponential equation used by many scientists to describe growth or decay events is:

This equation is very similar to y=ab^x, the equation introduced in our module Exponential Equations in Science I: Growth and Decay. In fact, we can map each component from one equation to the other:

• N₀ is the amount of something at time 0, which is the same as the initial value a.
• e is a constant (of approximate value 2.71828) that replaces the base value b.
• k is a constant that determines how quickly the value grows or decays, called the growth or decay rate constant.
• t is the variable of time, which replaces the variable x.
• N is the amount of something, equivalent to the variable y, which depends on the initial value, the growth rate, and time.

Note that k and t are multiplied by each other in the equation. Because k is a rate, its units are "per unit time," and might be per year (yr^-1) or per hour (hr^-1). The variable t has the units of time: year, or hour. When k is multiplied by t, therefore, their units cancel and we are left with a unitless exponent. You will see examples of this later in the module.

Where do these constants e and k come from? And why are they present in so many exponential equations that are used in science? First of all, it is important to point out that e and k are very different kinds of constants. Specifically, k is a constant whose value differs for each material or process (for example, the k-value for decay of ¹⁴C, a radioactive isotope that decays to ¹⁴N, is different than the k-value for ²³⁸U, another radioactive isotope that decays to ²⁰⁶Pb). In contrast, e is always e; it always has the exact same value. But what is that value, and why does it show up in exponential equations?

The constant e

You might have learned about the number e before: It is one of the most commonly used irrational numbers, which are numbers that cannot be expressed as fractions. The first 32 digits of e are 2.7182818284590452353602874713527. But that’s only the first 32 digits—in 2010, Shigeru Kondo succeeded in calculating the value of e to 1 trillion digits (Yee, 2011).

But where did e come from, and what does it mean? Throughout the 1600s, many mathematicians in Europe were working with exponents, exploring both the mathematical concepts and the applications of those concepts in everything from astronomy to finance. In 1683, the Swiss mathematician Jacob Bernoulli was studying the expression (1 + 1/n)ⁿ. He recognized that this expression was involved in the calculation of compound interest, an important financial topic since Sumerian merchants recorded interest calculations on clay tablets as early as 1700 BCE (Maor, 1994) (see Figure 1 for an example).

Interest is what a bank (or other entity) will pay you on your investment, usually given in the form of an interest rate, like 5% per year. Let’s say you invested $100 in a savings account at your bank at a 5% interest rate. If your bank were to calculate the amount to pay you using simple interest, they would pay you 5% per year on your initial investment, so you would get 5% of $100, or $5, every year. If they use compound interest, however, they calculate your interest on the total amount in your account. So the first year, you would make $5. The second year, you would make 5% interest on $105, or $5.25. The third year, you would make 5% interest on $110.25, or $5.51. In other words, you would make a little bit more every year.

Bernoulli was exploring the idea of paying out that interest more frequently: twice a year, or three times a year. In other words, if you wanted to maximize the interest you were paid on your investment, how frequently should that interest be calculated? In Bernoulli’s expression ${(1+\frac{1}{n})}^n$, the number n refers to the number of times per year that the interest was calculated on your investment. As he worked with this expression, Bernoulli observed that, as n became larger and larger, the value of this expression always fell between 2 and 3, even for very large n values (see Table 1).

Looking at the values in Table 1, you can see that while the values of the expression are always increasing, the amount of the increase in the solution to the equation gets smaller and smaller as n increases. If you think about this in terms of calculating interest, Bernoulli’s result shows that you don’t gain much more from calculating and paying out the interest ten times a year than you get from calculating it six times a year, and the biggest difference comes between one and three times a year. As larger and larger values of n are substituted, the value of the expression approaches a constant, which is approximately 2.71828—the value reported by Bernoulli as the limit of the expression (Figure 2). This would be the value of the interest paid on your investment if you had a 100% interest rate and if it were paid out constantly, an infinite number of times per year. The value of this limit is the number that we call e today. While Bernoulli formulated the expression and found the value, another mathematician, Leonard Euler, is credited with formalizing the constant with the designation e in the early 1700s.

So e is the limit to Bernoulli’s equation, but why is it used so commonly in exponential equations in science, replacing the base value b? What application does this have beyond calculating interest? To address those questions, it can be helpful to return to an exponential equation such as y = 2^x. Table 2 shows a series of y values for that equation.

If each of the values of x is considered to be a time step of equal length (such as an hour, a day, or a year), then with each time step there is a doubling of y. But the exponential equation y = 2^x assumes that all of that doubling occurs right at the time step, rather than gradually. For example, if you had 100 bacterial cells sitting in a petri dish and you knew that they split about once an hour, using this equation would mean that all 100 of those cells would wait 60 minutes and then all split at the same time. That is not realistic, however. Instead, each bacterium splits at a different time, but after an hour has passed, you would expect them all to have split – and some of the bacteria that had split at the beginning of the hour would be starting to split again. In other words, the population of bacteria is continuously growing over that hour, rather than doubling all at once – their growth follows the same mathematical progression as compound interest at an interest rate of 100%, assuming that the interest were being paid out continuously, as we described above.

There is also a limit to how quickly that population can grow. A single bacterium can only split into two, not into four or five. This natural, continuously and gradually growing population is therefore not accurately described by an exponential equation with a base value (b) of 2, but by an exponential equation with a base value (b) of e. The constant e has many uses in math, but in science we consider it the base rate of change (growth or decay) in systems that are continuously changing (growing or decaying) over time. Those systems might be a population of bacteria, a chemical reaction, or the decay of a radioactive material.

In science, the constant e is intimately connected to the constant k, often called the growth rate constant. As an exponent in the equation N = N₀e^kt, k modifies the base rate e for a specific material, process, or environmental condition. For example, most bacteria split more rapidly at room temperature than at near freezing, so the same bacteria will have a different growth rate constant k at different temperatures since there are different environmental conditions. Both are growth rates, so in both cases k is positive. By comparison, two different radioactive isotopes, such as ¹⁴C and ²³⁸U , have different values of k even though they both decay radioactively because they are different materials. But because the two isotopes are decaying, or decreasing over time, both growth rate constants are negative – a different process than reproduction.

Solving exponential equations that use e

The equation N = N₀e^kt can be solved for any of the variables within it, and sometimes scientists are interested in finding t (the amount of time that has passed), k (the growth or decay rate of something in a given set of conditions), N₀ (the initial amount of something), or N (the amount of something after a known amount of time has passed). The following sample problems address some of these different possibilities.

Sample Problem 1: Bacterial growth

We are often advised to refrigerate foods after opening them. Leftovers that sit outside the refrigerator overnight might “go bad” quickly even though they will last for several days in the refrigerator. Why is this the case? One reason is the presence of bacteria—bacteria are everywhere, and some are good for us and some are bad for us. Bacteria have been shown to display an exponential growth rate in many cases, but the rate of growth (or the steepness of the curve on the graph) varies significantly depending on temperature.

A study published in 1991 tested growth rates of the bacterium Lactobacillus plantarum (Figure 3), a generally benign bacterium found in fermented foods, at several different temperatures ranging from 6° C to 43° C (Zwietering et al., 1991). After allowing cultures to grow for 24 hours, the authors of the study determined the growth constant (k) for these different temperatures. At 6.0° C (about what you would find in a refrigerator), they found the growth rate constant of k = 0.0164 hr^-1, while at 28° C (close to room temperature), they found a growth rate constant of k = 0.8 hr^-1. (Remember that the units for k are “per unit time”, and that k can differ by material, process, or, as in this case, environmental variables like temperature.) What does this mean for the number of L. plantarum that would grow in your yogurt if you left it out of the refrigerator overnight for 12 hours?

For reference, here is a key to the exponential equation, N = N₀e^kt, for this problem:

N = total number of L. plantarum bacteria

N₀ = initial amount of L. plantarum bacteria

k = L. plantarum growth rate

t = time passed

e = a constant, approximately 2.71828

Using the equation N = N₀e^kt, we can determine the number of bacteria at both temperatures after 12 hours. First, let’s figure out how many bacteria there would be if we put the yogurt back into the refrigerator at 6° C. We know that there are some L. plantarum already in the yogurt, so we can assume the following values:

N₀ = 10 (we'll assume this initial value, though it's likely much higher)

k = 0.0164 hr^-1

t = 12 hr

Substituting these values into the equation, we get:

N = N₀e^kt

N = 10 e ^0.0164(12)

N = 12

As you can see, the number of bacteria does not increase very much in the refrigerator. Let’s try the same equation at the higher temperature, where k = 0.8 h^-1:

N = N₀e^kt

N = 10 e ^0.8(12)

N = 147,648 or N = 1.5 x 10⁵

That’s a whole lot more bacteria than you started with – what appears to be a very small change in the value of k results in a very large change in N. This is a characteristic of exponential equations, described in more detail in our module Exponential Equations in Science I: Growth and Decay.

Sample Problem 2: Carbon-14 dating

Carbon-14 dating (or ¹⁴C dating) is a technique that can be used to determine the age of anything that was once living that has carbon in it, from pieces of wood and charcoal to bone and skin. ¹⁴C is a radioactive isotope present in small amounts in the atmosphere and all living organisms, and it decays exponentially with a decay rate of k = -0.00012 yr^-1. While an organism is alive, it is constantly exchanging ¹⁴C with the atmosphere, replenishing the decaying isotopes and maintaining an approximately constant amount. Once an organism dies, however, that exchange and replenishment stops, and the ¹⁴C that was present at the time of death simply decays. (Note that because this is an exponential decay process, the value of k is negative. In earlier sample problems, we looked at growth processes, and k was positive.) Because we know the decay rate, we can determine how old something is if we know how much ¹⁴C is left from the starting amount (for more information about how ¹⁴C dating works, see our module on Uncertainty, Error, and Confidence).

In 1995, scientists from the University of Arizona, the US Department of Energy’s Brookhaven National Laboratory, and the Smithsonian Institution used ¹⁴C dating to determine the age of a controversial parchment that might be the first-ever map of North America, the Vinland map (shown in Figure 4). Text on the map reads, in part:

By God's will, after a long voyage from the island of Greenland to the south toward the most distant remaining parts of the western ocean sea, sailing southward amidst the ice, the companions Bjarni and Leif Eiriksson discovered a new land, extremely fertile and even having vines, ... which island they named Vinland.

This map, if authentic, suggests that the Vikings knew of North America before Columbus.

The scientists cut a small strip of parchment from the Vinland Map’s lower right corner and ran different parts of the sample through the dating process (Donahue et al., 2002). After running numerous tests the group found that the amount of ¹⁴C in the paper was approximately 93.5% of the modern value. Using this information and the equation N = N₀e^kt, they had what they needed to find the age of the parchment. We know that:

N = N₀e^kt

N₀ = 100% or 1.0

k = -0.00012 yr^-1

Substituting these values, we get:

N = N₀e^kt

0.935 = 1 e ^-0.00012t

In order to solve for t, we use an operation called the logarithm, and take the natural logarithm (ln) of both sides.

ln0.935 = -0.00012t

-0.0672 = -0.00012t

t = -0.0672 / -0.00012

t = 560 years

According to our calculations, the map was 560 years old in 1995. This suggests that the map was drawn around the year 1435, 50 years before Columbus came to America, suggesting that the Vinland map is authentic.

Applications of exponential equations in science

Equations in the form of N = N₀e^kt are ubiquitous in science. They are used in Earth science to determine the age of rocks based on the radioactive decay of long-lived isotopes such as ⁴⁰K, in epidemiology to predict the growth and spread of a virus with a particular incubation period, in chemistry to describe reaction rates, in populations of organisms like rabbits to manage their population, and in countless other ways. All of these systems exhibit not just a simple doubling or tripling (or halving) over time, but are continuously growing or decaying, and are thus more accurately described by a base growth rate of e rather than an integer.