Exponential functions: First a riddle

Here's a riddle: I'm offering you a job. It's a nasty one, like mucking out chicken coops (big ones, hundreds of chickens – and "mucking out" means shoveling ammonia-smelling chicken poop) 12 hours a day, for 30 days — no lunch breaks. At the end of the first day, I'll pay you one cent. One cent. At the end of the second, I'll double that and give you two cents, and so on for 30 days.

Think it through. On the seventh day, I'll pay you 64 cents, and you will have done 84 total hours of wretched labor for a grand total of \$1.27. Congratulations on your great decision and awesome survival instinct. Ready to quit yet?

Skip ahead to day 14. At the end of the day, I'll pay you \$81.92. Now that's more like it. And your cumulative total is now up to \$163.83 ... better, and remember that tomorrow your wage will double to \$163.84 for one day's work. Would you take the job?

In the table are the daily payments and cumulative earnings for just the last 10 days of your employment in the coops.

On the last day alone, I'd have to pay you over 5 million dollars! All-told, you'd earn more than \$10.7 million! Take the job!

Form of the function and its graph

The most important feature of an exponential function is that the independent variable is in the exponent of some base, usually an agreed-upon (but constant) base, like 2, 10 or the special number "e".

The graphs of exponential functions are characterized by a period of relatively slow growth followed by much more rapid growth, overtaking polynomial functions of even the highest degree.

Exponential functions

where b is the base and the independent variable (x) is in the exponent.

Rapid growth

In this illustration, the functions $f(x) = x^2$ (red) and $f(x) = 2^x$ (black) are plotted together. Although these curves are very similar over a small range (they are equal at $x = 2$ and $x = 4$), they are very different outside of it. Moving from left to right, exponential growth always outpaces polynomial growth, if $x$ gets large enough.

Notice the difference in the two functions when $x \lt 0.$ Recall from the laws of exponents that $a^{-1}$ means to take the reciprocal of a. So as x grows in the negative direction, the function asymptotically approaches zero: ½, ⅓, ¼, ...

The value of an exponential functions is never less than zero unless the function is specifically multiplied by a negative number - a vertical scaling parameter. Remember that a negative exponent does not make the number negative: $a^{-1} = 1/a.$

The graph of an exponential function

Exponential graphs all have a common shape, and sketching them is pretty easy. Check out the graph of $f(x) = 2^x$ on the right.

Without any transformations (covered below), every exponential function $f(x) = b^x,$ where b is a constant base (2 in this case), passes through y = 1, because anything raised to the zero power is 1: b⁰ = 1

On the right we have the rapid exponential growth. On the left, as the exponent approaches -∞, the function approaches zero. Remember that a negative exponent like b^-1 doesn't mean a negative number; it means 1/b, so as x → ∞, the value of the function approaches zero, thus we have asymptotic behavior on the left.

Transformations

All of the usual transformations of functions apply to exponential functions (below). One of the most important parts of an exponential function is the base. While any exponential phenomenon can be modeled using any base at all (by adjusting the rest of the transformation parameters to fit the data), we tend to stick to a small set of agreed-upon bases, like 2, 10 and e. There will be much more to say about bases, especially e, later.

Remember that all of the transformations work in the same way on any function. Things are always more the same in algebra than different.

Simple exponential growth

Consider the growth of money deposited in a bank account earning some annual interest rate, r%. (We use the letter r rather than a number so that our result will be good for any number.) One of the first challenges in solving problems with exponential functions is learning the new language. "Annual interest" means that every year, what's in the bank account is multiplied by the interest rate, r, and that amount is added to what's already there (the principal).

If an initial amount, A_o is deposited in a bank account earning r% interest, and not touched afterward, interest is earned each year. But after the first year interest is also earned on previous interest – that's compound interest

Example 1 — simple growth

$100 is deposited in a savings account that pays 5% interest per year. If no additional money is added, how much money will be in the account at the end of the 10^th year?

Solution: The phrase "annual interest" is a clue that this is a simple growth problem. Later we'll encounter compound growth – more on that later. The best approach is to start with the simple growth formula, and identify the things we know:

Example 2 – simple growth

If an initial investment of \$1000 earned $185 in interest after 8 years, what was the average rate of interest over that time?

Solution: Take the same approach for this problem; write the simple growth equation and identify what we know. The one trick here is that if we earn \$185 in interest on a $1000 investment, the total amount in the account after 8 years is A(8) = \$1185, the sum of the initial investment and the interest.

Now the variable for which we must solve is inside a binomial raised to the 8^th power:

Example 3 – simple growth - E. coli

At the right temperature (37˚C, the temperature of your gut) and with enough nutrients, E. coli populations will double every 20 minutes. Starting with an initial population of one bacterium, with time, t, in units of 20 min., How many bacteria will be present in 8 hours?

Solution: This problem has a couple of tricks you'll need to know. First, we have the word "double", which has a definite meaning in the simple growth equation, and second, we'll have to use "doubling" to calculate the growth rate before we can calculate the final population of E. coli. Finally, we're going to let 1 unit of time equal 20 minutes in this problem. It's not necessary at all, but it does simplify the arithmetic.

Start with the simple growth equation (I've used P for "population.")

Now we can solve for the rate. Not surprisingly, using this time scale, it's 100% over 20 minutes – the population doubles.

$$ \begin{align} 2 &= 1 (1 + r)^1 \\[5pt] r &= 1 \end{align}$$

100% growth every 20 min.

Now we can use r to construct our growth formula:

$$P(t) = P_o \cdot 2^t$$

Example 4 — depreciation

Let's say you own a pizza restaurant and you buy a new oven. Your business is now worth more and will therefore be taxed accordingly. But the government (you and me) recognizes that the value of the oven will decrease over time, and provides a way to compensate you for that. It's called depreciation. Such an oven might depreciate by 5% per year. Calculate the taxable value of an oven originally purchased for \$10,000 after 10 years with a 5% depreciation rate.

Solution: This is really just a simple growth problem, except that the growth is negative, so instead of the function A(t) = A_o(1 + r)^t, we use the depreciation form:

We start by writing the depreciation equation and what we know:

Practice problems — simple growth

Compound exponential growth

Keeping with our bank-account example, what if, instead of adding the interest to the balance at the end of each year, we added it in chunks periodically throughout the year. In banking, that's called compounding the interest. The idea will extend far beyond banking, so it's worth studying.

Now, of what value is this to a bank customer (and of what value is compounding in the mathematics of exponential growth)? Take a look at the figure below to see how.

Let's say we deposit \$1000 into an account that pays 4% interest per year, and we wait 10 years. According to the simple interest formula, we'd end up with about \$1480. The graph shows that there is about a \$9 advantage in switching to compounding quarterly (4 times a year).

Notice that as we increase the frequency of compounding, the amount of money gained in compounding more often (the "marginal gain") is measurable, but less. The dashed line is an asymptote.

That asymptote represents the limit as compounding becomes continuous, and that is the essence of the next section.

Practice problems — compound growth

What if the number of compounding steps is huge?

If you take a look at the compound interest graph above, you'll notice that there is a limit to the amount of money that can be made by increasing the number of compounding steps. The word "steps" here is the key.

With compound growth, we wait a while, then add interest, then wait a while more and add interest, and so on. It's a stair-step process. But what if we took steps so frequently that the process was more like a ramp than a stairway?

That would be continuous growth, and the problem we'd be trying to solve is this one:

$e$ on your calculator

Example 5 — compound interest

Compare the amount of money in a bank account after ten years at an interest rate of 4% per year, when that interest is compounded annually and quarterly. Assume that \$10,000 is initially deposited and that no more deposits are made.

The first part of the problem is a simple growth problem, so we'll write that function and what label what we know:

Now it's easy to solve:

$$ \begin{align} A(t) &= 10,000(1 + 0.04)^{10} \\[5pt] &= 10,000 (1.04)^{10} \\[5pt] &= 10,000(1.4802) \\[5pt] &= \$ \, 14,802 \end{align}$$

The second part is a compound-growth problem with n = 4 (quarterly):

Example 6 – More compound interest

Compare the interest earned after ten years in a savings account started with \$1000.00 (no additional deposits) if the 5% interest is compounded quarterly (4 times/year), weekly (52 times/year), daily and continuously.

Quarterly:

$$ \begin{align} A(t) &= 10,000 \left( 1 + \frac{0.05}{4} \right)^{4 \cdot 10} \\[5pt] &= 10,000 (1 + 0.0125)^{40} \\[5pt] &= 10,000(1.0125)^{40} \\[5pt] &= 10,000(1.6436) \\[5pt] &= \$ \, 16,436 \end{align}$$

Weekly:

$$ \begin{align} A(t) &= 10,000 \left( 1 + \frac{0.05}{52} \right)^{52 \cdot 10} \\[5pt] &= 10,000 (1 + 0.000962)^{520} \\[5pt] &= 10,000(1.000962)^{520} \\[5pt] &= 10,000(1.6483) \\[5pt] &= \$ \, 16,483 \end{align}$$

Daily:

$$ \begin{align} A(t) &= 10,000 \left( 1 + \frac{0.05}{365} \right)^{365 \cdot 10} \\[5pt] &= 10,000 (1 + 0.000137)^{3650} \\[5pt] &= 10,000(1.000137)^{3650} \\[5pt] &= 10,000(1.6487) \\[5pt] &= \$ \, 16,487 \end{align}$$

Example 7 – Population growth

A population of seabirds has grown reliably at a rate of 4% per year. If the population today is 16,000 birds, what was it 5 years ago?

Solution: This is a continuous growth problem because it says so – watch for that! Notice also that we never actually know anything about the actual bird population. We just have changes, but that will be fine.

We begin by writing the continuous-growth function and sketching in things we know

Practice problems — continuous growth

Half-life – Another way to think of exponential decay

In problems that involve exponential decay, it is conventional to use the half-life measure. Here's how it works. Think about a pile of Iodine-131 (¹³¹I). This radioactive isotope decays spontaneously to ¹³¹Xe by emission of a beta particle followed by some gamma radiation. It's not possible to tell when a single atom of ¹³¹I will decay, but we know that on average for a large enough quantity, one half of the initial amount will have decayed to ¹³¹Xe in 8.02 days, the half-life of ¹³¹I. Every 8.02 days, half of what remained will have decayed and become ¹³¹Xe.

It's not important that you know the chemistry or physics of these problems right now, just that at any given time, what's left will be cut in half (turn to something else) in about 8 days.

The half-life function

To derive a formula for finding the amount of a radioactive substance still left after some number of half lives, we start with the formula for simple growth:

$$A(t) = A_o (1 + r)^t$$

Now if 1/2 of the sample is removed every time unit, then r = -0.5, so we have

$$A(t) = A_o \left( 1 - \frac{1}{2} \right)^t$$

Example 8 — Radioactive decay & half life

If the half life of a substance is 100 days, how much of the initial sample will be present after one year?

Solution: Notice that we haven't been given an initial amount (A_o) here. That's OK because the answer, expressed as a fraction (percent) of the starting amount, would be the same for any starting amount. For convenience, let's say we started with 1 (gram, Kilogram, whatever). Plugging in k = 100 days, t = 365 days and A_o = 1, we have:

Video examples

Some of these videos involve using logarithms to solve exponential problems. Don't worry about those until you learn about "logs," but those are crucial problems to learn, so do come back.

1. Deriving the formula for simple exponential growth

In this example, we derive the formula for simple exponential growth, A(t) = A_o(1 + r)^t, from scratch by considering a bank account that pays r% interest per year. The trickiest thing about this definition is factoring things out of an expression. Remember that x + 2x = x(1 + 2). It's a simple pattern that's applied in this derivation exactly as shown, just with more complicated things in the places of the x's , 1's nad 2's.

Minutes of your life: 3:22

3. Finding a continuous-growth model from data

Given two data points, P(1) = 0.1 million and P(7) = 1.2 million, can we find a continuous-growth model, P(t) = P_oe^rt ? This really amounts to solving a two-equations, two-unknowns problem with logs.

Minutes of your life: 2:14

5. Radioactive decay of plutonium-239

Nuclear power plants generate ²³⁹Pu, which is both toxic and highly radioactive. The half life of ²³⁹Pu is 24,000 years. In this example we'll calculate how long it takes for any sample of ²³⁹Pu to decay to 10% of its original mass. The answer is shocking!

Minutes of your life: 2:52

7. Depreciation

Things lose value over time, and we need a way to estimate the value of an object with the passing of time. Think about buying a new car. After a couple of years, it's a used car. You couldn't possibly sell it for what you originally paid. We call that loss of value depreciation, and we model it with the simple growth formula, A(t) = A_o(1 + r)^t, but where the rate r is negative.

Minutes of your life: 1:58

9. Solving logarithmic equations (1) – two examples

In this video, we solve two logarithmic equations, log(x) + log(x + 3) = log(7) and 2 ln(x) = ln(12x - 27). We use the laws of logs to re-express each so that the solutions are easy to get with simple algebra.

Minutes of your life: 4:15

11. Graphing an exponential function & its inverse (1 of 2)

The key to graphing exponential and logarithmic functions is remembering that they're inverses, and have mirror symmetry across the line y = x. Here we sketch the graph of y = e^x, and its inverse, y = ln(x).

Minutes of your life: 2:48

13. The strangest of the laws of logs:

log_b(x^a) = a·log_b(x)

Here are some manipulations of log and exponential functions that might help you come to terms with this, the most useful (and also most confusing – at first) law of logs.

Minutes of your life: 2:04