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-smelly 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!

 

That's exponential growth, and you'll see it in many different phenomena, like financial growth (compound interest) and population growth. Its cousin, exponential decay is used to model things like radioactive decay, the decrease in drug concentration in the bloodstream over time and the depreciation (loss of value) of property over time. And there are many other examples.

Knowing how to manipulate and solve exponential functions and equations is a very important part of your mathematics toolkit.


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 are different

In an exponential function, the variable is in the exponent. The base is constant.

Rapid growth

In this illustration, the functions f(x) = x2 (black) and f(x) = 2x (orange) 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 < 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) = 2x on the right.

Without any transformations (covered below), every exponential function f(x) = bx, 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.

 


Simulation

Use the sliders on the graph of   f(x) = bx+h + k   below to get a feel for the effect on the graph of changing the vertical and horizontal translation parameters h and k, and how changing the base changes the graph.

Simple 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).

For every year after the initial deposit, this means that you're calculating this year's interest based not only on the initial amount deposited, but on last year's interest as well. That's called compound growth or compound interest.

Try to work your way through the table below. The algebra is pretty straightforward - just finding finding common factors between terms.

Amount in account each year at an annual interest rate of r%

If an initial amount, Ao 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

Year Amount Condensed
0 Ao (initial investment) Ao(1 + r)0
1 Ao + Aor = Ao(1 + r) Ao(1 + r)1
2 Ao(1 + r) + r·Ao(1 + r) = Ao(1 + r)(1 + r) = Ao(1 + r)2 Ao(1 + r)2
3 Ao(1 + r)2 + r·Ao(1 + r)2 = Ao(1 + r)3 Ao(1 + r)3
. . .    
n years Ao(1 + r)n (the pattern continues) Ao(1 + r)n

Simple growth formula

For a system that grows at a rate, r, added periodically over t periods, the amount A(t) at any time t is

A(t) = Ao (1 + r)t

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 10th 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:

Now it's just a matter of completing the calculation to find A(10), the amount in the account after t = 10 years.

That's it. Just remember to convert the percent interest to decimal form: 5% = 5/100ths, or 0.05.


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 8th power:

To isolate r, we have to get rid of the 8th power, and we do that by raising both sides of the equation to the ⅛ power (remember the laws of exponents).

So an interest (growth) rate of 2.14% per year would do the trick.


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.

Now we can use r to construct our growth formula:

E. coli bacteria magnified 15,000 times

e coli

Noting that 8 hours is 24 of our 20 min. periods,

We can calculate the number of bacteria.

17 million! Exponential growth is wild!


Compound 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.

For example, let's say we have a bank account that pays 4% annual interest, and we do what banks call "compounding quarterly." That is, we split the 4% up into four equal pieces (1%) and pay that 1% in interest every three months. This begs two questions: (1) is there an easy way to do it, and (2) is it to our advantage?

To develop a formula for compounding, we can start with the simple growth formula and just modify it:


 


Compound growth formula

For a system that grows at a rate, r, compounded n times in each of t periods, the amount A(t) at any time t is

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.

Note that the horizontal axis of this graph has no real scale; the data are just spread evenly across it to make it easy to read.

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:

It's a tricky process to go through, but the answer to this question turns out to be very elegant. This equation, in the limit where n → ∞, converts to

where e is a very special number, the base of all continuously-growing exponential functions, e = 2.7182818 ... It's a transcendental number, like π, and you'll probably use it a lot, so get used to it.

We'll use this new function for continuous growth to model many types of exponential growth situations, especially populations. When populations become large enough, and births and deaths occur more-or-less simultaneously, the growth is approximately continuous, and the P(t) = Poert is a good model.

If you want to know more about where e comes from, go here:

Continuous growth formula

For a system that grows continuously at a rate, r, the amount A(t) at any time t is

A(t) = Aoert


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) = Ao(1 + r)t, we use the depreciation form:

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

Then it's just a matter of finishing the calculation:

After ten years the pizza oven is worth a little more than half of its original value.


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.

Solution

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:

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

And solving that gives us:

The compound interest option gives us more money at the end of ten years. Cool.


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.

Solution

Quarterly:

Weekly:

Daily:

Continuously – here let's pause to write our function and plug in values more carefully:

and the result is:

Notice that the final amounts increase, but by less and less as we add more compounding steps, and that the difference between daily compounding and continuous growth is less than a cent, but the continuous case is still just a little greater – check it out for yourself.


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

Now let's rearrange the function before plugging in numbers (always a good idea):

Now plug in what we know

and the result is

Continuous growth models like this are almost always used to model populations.


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.

Look at the graph to see how it works. It's pretty clear that this is a upside-down exponential graph.

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:

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

Then we can introduce the half life, which we'll call k, to put the exponent in units of number of half lifes:

It's that exponent that's usually a little tricky. Think of it this way: If the half life is 8 days and we wait for 16 days, so t/k = 16/8 = 2 half lifes. We divide by k to convert the exponent to half lifes.

The half-life function

For exponential decay expressed in half-lifes, the function is:

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 (Ao) 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 Ao = 1, we have:

Now it's just a matter of plugging in and solving for A(365).

So after one year, only 7.9% of the isotope will be present. The rest will have decayed to other stuff.


Practice problems

  1. You deposit $2,300 in a bank account. Find the balance after four years for each of these situations:
    1. The account pays 3.2% annual interest, compounded quarterly.
    2. The account pays 2.22% annual interest, compounded monthly.
    3. The account pays 2.00% annual interest, compounded daily.

  2. Let's say you would like to have $3,500 in your savings account three years from now. Find the amount you should deposit today for each of the account types below in order to acheive your goal.
    1. The account pays 3.35% interest compounded quarterly.
    2. The account pays 4% annual interest.
    3. The account pays 2.25% annual interest compounded monthly.

  3. A couple retiring today will need to have about $2 million invested in order to retire comfortably and afford health care costs. When you retire that amount will likely have doubled about twice because of inflation. Assuming you'll need to have about $8 million invested by the time you retire, how much money would you need to put in the bank right now in order to retire at age 65? (Of course, this is a bit unrealistic because you'll be contributing as you go, but it might still be instructive.) Assume a 6.5% interest rate over the whole time.

  4. Let's say you buy a new car for $30,000. If the annual depreciation rate is 12%, how much will the car be worth in 6 years? (Remember that a 12% depreciation rate can be seen as a -12% rate of "growth").

Video examples


1. Deriving the formula for simple exponential growth

In this example, we derive the formula for simple exponential growth, A(t) = Ao(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

2. Interest earned: Simple vs. Compound

In this example, we compare the interest on $1000.00 deposited for ten years (no other deposits) at 8% annual interest, A(t) = Ao(1 + r)t and at 8% annual interest compounded quarterly: A(t) = Ao(1 + r/n)nt.






Minutes of your life: 2:29


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) = Poert ? This really amounts to solving a two-equations, two-unknowns problem with logs.




Minutes of your life: 2:14

4. Dating of old materials by 14C decay: A half-life problem

When plants and animals are alive, they exchange carbon atoms with the environment continually. When they die, that stops and the radioactive 14C (one in every trillion carbon atoms) begins to decay. That decay can be measured. Using the half-life formula, A(t) = Ao(1/2)t/k and the half life of 14C (5730 years) we can calculate the age of an artifact.

Minutes of your life: 3:14


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

6. Using logs to calculate time in a simple growth problem

This is a simple growth problem using A(t) = Ao(1 + r)t, but this time we use logarithms to solve for t, the time. Whenever we need to solve for a variable in an exponent, we have to use logarithms. Logs are the inverses of exponential functions. If f(x) = 10x, then f-1(x) = log10(x). Recall that f-1(f(x)) = f(f-1(x)) = x. The actions of a function and its inverse on one another "cancel."

Minutes of your life: 2:19


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) = Ao(1 + r)t, but where the rate r is negative.

Minutes of your life: 1:58

8. Derivation of the change of base formula

How do we calculate log382 ? 82 isn't a nice power of 3 (like 81 is), so it's difficult. Most calculators have a way to plug in logs with bases that aren't 10 (common logs) or e (natural logs), but you don't really need those. There's an easier way to do it and it's called the change of base formula.


Minutes of your life: 3:26


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

10. Solving logarithmic equations (1) – two examples

Here we solve two logarithmic equations where the key is to recognize a common base and reexpress everything in terms of that base: log16 = 1/4 and log3 = 81


Minutes of your life: 2:04


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 = ex, and its inverse, y = ln(x).

Minutes of your life: 2:48

12. Graphing and exponential function & its inverse (2 of 2)

Sketch the graph of an exponential function that has been transformed a bit: f(x) = 3x-2. Then calculate and sketch its inverse on the same axes.


Minutes of your life: 2:55


13. The strangest of the laws of logs: logb(xa) = a·logb(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

 

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