The exponential function is one of the most important functions in mathematics (though it would have to admit that the linear function ranks even higher in importance). To form an exponential function, we let the independent variable be the exponent. A simple example is the function
$$f(x)=2^x.$$
As illustrated in the above graph of $f$, the exponential function increases rapidly. Exponential functions are solutions to the simplest types of dynamical systems. For example, an exponential function arises in simple models of bacteria growth
An exponential function can describe growth or decay. The function
$$g(x)=left(frac{1}{2}
ight)^x$$
is an example of exponential decay. It gets rapidly smaller as $x$ increases, as illustrated by its graph.
In the exponential growth of $f(x)$, the function doubles every time you add one to its input $x$. In the exponential decay of $g(x)$, the function shrinks in half every time you add one to its input $x$. The presence of this doubling time or halflife is characteristic of exponential functions, indicating how fast they grow or decay.
Parameters of the exponential function
As with any function, the action of an exponential function $f(x)$ can be captured by the function machine metaphor that takes inputs $x$ and transforms them into the outputs $f(x)$.
The function machine metaphor is useful for introducing parameters into a function. The above exponential functions $f(x)$ and $g(x)$ are two different functions, but they differ only by the change in the base of the exponentiation from 2 to 1/2. We could capture both functions using a single function machine but dials to represent parameters influencing how the machine works.
We could represent the base of the exponentiation by a parameter $b$. Then, we could write $f$ as a function with a single parameter (a function machine with a single dial):
$$f(x)=b^{x}.
$$
When $b=2$, we have our original exponential growth function $f(x)$, and when $b=frac{1}{2}$, this same $f$ turns into our original exponential decay function $g(x)$.
We could think of a function with a parameter as representing a whole family of functions, with one function for each value of the parameter.
We can also change the exponential function by including a constant in the exponent. For example, the function $$h(x)=2^{3x}$$
is also an exponential function. It just grows faster than $f(x)=2^x$ since $h(x)$ doubles every time you add only $1/3$ to its input $x$.
We can introduce another parameter $k$ into the definition of the exponential function, giving us two dials to play with. If we call this parameter $k$, we can write our exponential function $f$ as
$$f(x)=b^{kx}.
$$
You can explore the influence of both parameters $b$ and $k$ in the following applet.
The exponential function. The exponential function $f(x)=b^{kx}$ for base $b >0$ and constant $k$ is plotted in green. You can change the parameters $b$ and $k$ by typing new values in the corresponding boxes. It turns out the parameters $b$ and $k$ can change the function $f$ in the same way, so you really only need to change one of them to see all the different functions $f$.
To see how they do the same thing, you can click the “fix function” checkbox, which will fix the function $f(x)$. When that box is checked, if you change the parameters $b$ or $k$, the other parameter will change in a way to leave the function $f(x)$ unchanged. For the function $f(x)=b^{kx}$, the value $f(0)=1$ for all parameters.
To change the value of $f(0)$, you can allow scaling of the function by clicking the corresponding checkbox. Then, the function changes to $f(x)=c b^{kx}$ with an additional parameter $c$ that scales (multiplies) the whole function so that $f(0)=c$. You can change the value of $c$ by dragging the red point. You can change range of the $x$ and $y$axes buttons labeled $x+$, $x$, $y+$, and $y$.
Since $f(x)$ is always nonnegative, only the positive $y$axis is shown.
More information about applet.
It turns out that adding both parameters $b$ and $k$ to our definition of $f$ is really unnecessary. We can still get the full range of functions if we eliminate either $b$ or $k$. You can see this fact through the above applet. For example, you can see that the function $f(x)=3^{2x}$ ($k=2$, $b=3$) is exactly the same as the function $f(x)=9^x$ ($k=1$, $b=9$).
In fact, for any change you make to $k$, you can make a compensating change in $b$ to keep the function the same. To see this, check the “fix function” checkbox. Then, if you change either $b$ or $k$, the applet will automatically make a compensatory change in the other parameter to keep the function the same.
If you are curious why this is true, you can check out the calculation showing the two parameters are redundant.
Since it is silly to have both parameters $b$ and $k$, we will typically eliminate one of them. The easiest thing to do is eliminate $k$ and go back to the function
$$f(x)=b^x.$$
We will use this function a bit at first, changing the base $b$ to make the function grow or decay faster or slower.
However, once you start learning some calculus, you'll see that it is more natural to get rid of the base parameter $b$ and instead use the constant $k$ to make the function grow or decay faster or slower.
Except, we can't exactly get rid of the base $b$. If we set $b=1$, we'd have the boring function $f(x)=1$, or, if we set $b=0$, we'd have the even more boring function $f(x)=0$.
We need to choose some other value of $b$.
What is Exponential Growth?
This is the first post in a threepart series on exponential growth and doubling time – concepts that are important in not only mathematics courses, but life science and AP Environmental Science courses. The series will explore these concepts, describe the calculations involved, and provide educators with resources for teaching students about these important topics.
Before we dive in to the concept of exponential growth – consider this brainteaser from our student activity Population Riddles: A father complained that his son’s allowance of $5 per week was too much.
The son replied, “Okay, Dad. How about this? You give me a penny for the first day of the month, 2 cents for the second, 4 cents for the next, 8 cents for the next, and so on for every day of the month.
” The father readily consented.
Who was more clever, the father or son?
If you answered the son, you are correct! In fact, the son’s allowance on the 31st alone would total $10,737,418.24! In this riddle, students quickly learn that doubling a small number over and over soon means doubling larger numbers. This phenomenon is the driving power behind exponential growth.
Exponential growth is growth that increases by a constant proportion. In the allowance riddle, the son requested that his father double the dollar amount (or increase the amount by 100%) each day beginning at $0.
01, making it a perfect example of exponential growth. It easy to identify a population experiencing exponential growth when we graph the data. Most graphs will exhibit a strong Jshape – often referred to as the “J curve”.
If we look at a graph of the son’s daily allowance we can see the J curve take shape starting at day 26.
One of the best examples of exponential growth is observed in bacteria. It takes bacteria roughly an hour to reproduce through prokaryotic fission. If we placed 100 bacteria in an environment and recorded the population size each hour, we would observe exponential growth.
We would record 200 at the start of the second hour, 400 at the start of the third, 800 at the start of the fourth, and so on. Eventually, we would observe a leveling off in population size due to various resource and ecosystem constraints. This is an important observation. A population cannot grow exponentially forever.
Over time, it will exceed its carrying capacity – an important concept in the life and environmental sciences.
Resources for Teaching Students about Exponential Growth
Population Education has many resources for teaching students about exponential growth. Be sure to check out our website for more great lessons on exponential growth!
 Population Riddles: Riddles that help students conceptualize large number and understand the concepts of exponential growth and doubling time.
 World Population DVD: Our award winning video is a strong visual representation of human population growth over time.
Exponential Growth Series
Coronavirus and Exponential Growth [UPDATED 4202020]
Exponential Growth of Virus: Updated 4202020
In the article below, we discuss the exponential growth and eventual decline of the coronavirus pandemic.
Gerry Harp, who is a former director of SETI research at this Institute, noted that the plots made available to the public showing the number of new virus cases are generally presented on graphs with a linear vertical axis. But for something that’s growing exponentially, a logarithmic scale would be a better choice – certainly for scientists.
Harp collected data from the CDC on new virus cases in the U.S. up through April 19, and plotted them on a semilog graph (above.) You can see the bending towards a slower growth rate for new cases.
 This is a better way of displaying the data for an exponential process.
 — Seth Shostak
 By Seth Shostak, Senior Astronomer
The spread of the coronavirus will be exponential – which is bad. But its inevitable decline will also be exponential, which is good.
Astute readers of media will have noticed that in the last two years the use of the word “exponential” has suddenly become fashionable. In the majority of cases, a writer will use it to mean “a lot”, as in “streamed video content has grown exponentially.”
But of course, the word is historically grounded in algebra, and was first used by Descartes to describe, for example, something whose growth with time can be characterized as ta, where a is the exponent. A simple example would be Moore’s Law: If the number of transistors on a chip doubles every two years, that’s an exponentially increasing improvement.
In the case of the coronavirus, the growth in the number of infected persons will inevitably be exponential, at least for a while.
That’s because the rate of new infections clearly depends on the number of people who are already contagious. The resulting tally of the infected will increase very rapidly – as is typical of exponential growth.
Note that it’s not that the number is large, but only the behavior of the growth rate that merits the designation “exponential.”
And, as noted, that’s clearly bad. The accompanying graph, from Nature magazine, shows the total number of coronavirus cases in China (red curve).
It rises rapidly and for awhile is exponential – with a doubling time of about a week during early February. But a glance at the plot shows the number of cases leveling off by midFebruary.
If this were a graph of new cases rather than total cases, the curve would have turned over, and headed back down.
And since the number of new cases also depends on the number of infectious people (which declines as folks recover), that will also be exponential, but exponentially decreasing.
Bottom line; While the bad news grows rapidly, the good news will also evolve rapidly. So let’s just hope that by invoking shelteringinplace and other strategies, we can reduce the infection rate and cause the inevitable bellshape curve of the number that are sick to turn over sooner.
Exponential Growth and Decay – MathBitsNotebook(A2 – CCSS Math)
In Algebra 1, the following two function formulas were used to easily illustrate the concepts of growth and decay in applied situations. If a quantity grows by a fixed percent at regular intervals, the pattern can be depicted by these functions.


Remember that the original exponential formula was y = abx. You will notice that in these new growth and decay functions,
the b value (growth factor) has been replaced either by (1 + r) or by (1 – r).
The growth “rate” (r) is determined as b = 1 + r. The decay “rate” (r) is determined as b = 1 – r
a = initial value (the amount before measuring growth or decay) r = growth or decay rate (most often represented as a percentage and expressed as a decimal) x = number of time intervals that have passed 
Example 1: The population of HomeTown is 2016 was estimated to be 35,000 people with an annual rate of increase of 2.4%. a) What is the growth factor for HomeTown?
After one year the population would be 35,000 + 0.024(35000). By factoring, we have 35000(1 + 0.024) or 35000(1.024).
The growth factor is 1.024.
(Remember that growth factor is greater than 1.)
b) Write an equation to model future growth. y = abx = a(1.014)x = 35000(1.024)x c) Use the equation to estimate the population in 2020 to the nearest hundred people. y = 35000(1.024)4 ≈ 38,482.91 ≈ 38,500 
Most naturally occurring phenomena grow continuously
. For example, bacteria will continue to grow over a 24 hours period, producing new bacteria which will also grow. The bacteria do not wait until the end of the 24 hours, and then all reproduce at once.
The
exponential e
Exponential Growth
Some things grow at a consistent rate. Money or the descendants of mating rabbits, for example, can grow faster and faster as the total number itself gets bigger. When growth becomes more rapid in relation to the growing total number, then it is exponential.
Exponential growth is extremely powerful. One of the most important features of exponential growth is that, while it starts off slowly, it can result in enormous quantities fairly quickly – often in a way that is shocking.
There is a legend in which a wise man, who was promised an award by a king, asks the ruler to reward him by placing one grain of rice on the first square of a chessboard, two grains on the second square, four grains on the third and so forth.
Every square was to have double the number of grains as the previous square.
Exponential Growth of RiceThe king granted his request but soon realized that the rice required to fill the chessboard was more than existed in the entire kingdom and would cost him all of his assets.
The number of grains on any square reflects the following rule, or formula:
In this formula, k is the number of the square and N is the number of grains of rice on that square.
 If k = 1 (the first square), then N = 2^0, which equals 1.
 If k = 5 (the fifth square), then N = 2^4, which equals 16.
This is exponential growth because the exponent, or power, increases as we go from square to square.
Speed of Exponential Growth
What Is Exponential Growth?
Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.
For example, if a population of mice doubles every year starting with two in the first year, the population would be four in the second year, 16 in the third year, 256 in the fourth year, and so on.
The population is growing to the power of 2 each year in this case (i.e., exponentially).
 Exponential growth is a pattern of data that shows sharper increases over time.
 In finance, compounding creates exponential returns.
 Savings accounts with a compounding interest rate can show exponential growth.
In finance, compound returns cause exponential growth. The power of compounding is one of the most powerful forces in finance. This concept allows investors to create large sums with little initial capital. Savings accounts that carry a compound interest rate are common examples of exponential growth.
Assume you deposit $1,000 in an account that earns a guaranteed 10% rate of interest. If the account carries a simple interest rate, you will earn $100 per year. The amount of interest paid will not change as long as no additional deposits are made.
If the account carries a compound interest rate, however, you will earn interest on the cumulative account total. Each year, the lender will apply the interest rate to the sum of the initial deposit, along with any interest previously paid. In the first year, the interest earned is still 10% or $100.
In the second year, however, the 10% rate is applied to the new total of $1,100, yielding $110. With each subsequent year, the amount of interest paid grows, creating rapidly accelerating, or exponential, growth. After 30 years, with no other deposits required, your account would be worth $17,449.40.
On a chart, this curve starts slowly, remains nearly flat for a time before increasing swiftly to appear almost vertical. It follows the formula:
The current value, V, of an initial starting point subject to exponential growth can be determined by multiplying the starting value, S, by the sum of one plus the rate of interest, R, raised to the power of T, or the number of periods that have elapsed.
While exponential growth is often used in financial modeling, the reality is often more complicated. The application of exponential growth works well in the example of a savings account because the rate of interest is guaranteed and does not change over time. In most investments, this is not the case. For instance, stock market returns do not smoothly follow longterm averages each year.
What are exponential growth models? + Example
An exponential growth model describes what happens when you keep multiplying by the same number over and over again. It has many applications, particularly in the life sciences and in economics.
A simple exponential growth model would be a population that doubled every year. For example,
#y=A(2)^x#
where A is the initial population, x is the time in years, and y is the population after x number of years. Having x in the exponent causes the initial value ( A) to keep doubling as x increases.
That's an Algebra explanation, though, and this is being asked for a Calculus course. So now I have to skip ahead.
When you're dealing with things that grow exponentially, the number e (2.71828…) shows up, similar to how #pi# shows up whenever you talk about circles and trigonometry. Here's why, using some of the techniques you should have learned in your Calculus course.
 Exponential models describe situations where the rate of change of some thing is directly proportional to how much of that thing there is.
 In math terms:
 #dy/dt=ky#
 where #dy/dt# is the rate of change in an instant, y is the amount of the thing we're talking about at that instant, and k is the constant of proportionality.
 This can be solved in the given terms to produce a general exponential growth/decay model.
 We can solve the differential equation by first separating the variables.
 #dy/dt=ky#
 Multiply both sides by dt and divide both sides by y.
 #dy/y=k dt#
 Then we integrate both sides.
 #intdy/y=intk dt#
 Hopefully, you've learned how to antiderive #dy/y# and #kdt#.
 #lny=kt+C#
 2 quick thoughts:
1) We can drop the absolute value in this case. We're talking about a population of some sort, or money. We won't need negative numbers.
2) We only have to write #+C# on one side. Yes, every time you integrate, a #+C# should appear. But whatever those constants might be, it's easy enough to collect them both on one side, and call the result #+C#.
Now, we are not done solving a differentional equation until we have solved for y. So we have to get rid of the natural logarithm (#ln#) by making both sides of the equations exponents with e as the base.
 So:
 #lny=kt+C#
 #e^lny=e^(kt+C)#
 #y=e^(kt+C)#
 #y=e^(kt) e^C#
Now, since C is a constant, then e^C is just some other constant. We'll call that constant C. Now we have:
 #y=Ce^(kt)#
 This is your general exponential growth model.
 y is the amount of the thing we're talking about after t units of time have passed.
 C is the initial amount of the thing we're talking about, when #t=0#.
e is the base of natural logarithms, and is approximately 2.71828…
k is a specific constant to each situation. There are several ways to solve for it, depending on the situation. You can make a quick rule for k by solving for it.
 #k=(ln(y/C))/t#
 Or, in English: k is the natural log of the amount at t divided by the initial amount C, all divided by how long it took to get from C to y.
 t is time.
Depending on the specifics of the problem your facing, it might make the most sense to just use the equation above as a formula. Sometimes, it makes sense to take the initial conditions and set them up as integral, and solve the new differential equation with the specific conditions.
Sometimes we use slightly different versions of the above model (for example (Newton's Law of Cooling). Sometimes, the model looks very different from the above (as in logistic growth models). But the principle is the same.
To give a more detailed answer, I would need a more specific question.
Hope this helps.
Coronavirus is growing exponentially – here’s what that really means
You may have seen a version of the infographic (below) that explains the potential impact of social distancing. It nicely illustrates that reducing the total number of diseasespreading contacts each infected person has can have a dramatic effect on the total number of infections a short time later. The numbers rely on the mathematical concept of “exponential growth”.
The total number of infected people reduces dramatically after 60 days with relatively small changes to the reproduction number of the disease. Christian Yates, Author provided
Recently, Boris Johnson, the British prime minister, told the press that “it looks as though we’re now approaching the fast growth part of the upward curve. And without drastic action, cases could double every five or six days”. The consistent doubling of cases in a fixed period is the hallmark of exponential growth.
The number of new infections that a single infectious individual will cause during their infectious period is known as the basic reproduction number of a disease. This number is key to determining how widespread a disease will become.
For COVID19, early estimates of the basic reproduction number have it somewhere between 1.5 and 4. The infographic assumes a figure somewhere in the middle, at 2.5 infectious contacts per infectious individual.
If the reproduction number of a disease can be brought below one, then the spread will slow until the disease dies out. The revised infographic below shows the number of currently infected people. Reducing contact with others by 75% will bring the reproduction number below the critical level, allowing the number of infected people to decrease almost to zero in just two months.
The number of currently infected people after 60 days (assuming recovery after 10 days) changes with the reproduction number of the disease. Christian Yates, Author provided
However, the basic reproduction number of the current stage of the outbreak is way above one. This means that each newly infected person will pass on the disease to at least one more person, on average, and consequently the disease will take off exponentially.
What exponential growth is
But what, precisely, is exponential growth? The mathematical definition says that a quantity that increases with a rate proportional to its current size will grow exponentially.
This means that as the quantity increases so does that rate at which it grows.
The more infected people we have in the early stages of a disease outbreak, the more people they will infect and the more the cases will rise.
The total number of confirmed cases of COVID19 in the UK is increasing exponentially. Max Roser, Hannah Ritchie and Esteban OrtizOspina (2020) – 'Coronavirus Disease (COVID19) – Statistics and Research', CC BY
Other situations in which exponential growth plays a critical role range from pyramid schemes to nuclear weapons. In a pyramid scheme, each new investor invites two more recruits who in turn invite two more.
This rapid growth, at a rate proportional to the current number of members, inevitably leads to a situation in which there aren’t enough new recruits to keep the scheme going and it eventually collapses.
When the pyramid tumbles, most investors lose their money.
In a nuclear fission bomb, a single uranium atom splits in two, jettisoning fastmoving neutrons and large quantities of energy in the form of electromagnetic radiation. The neutrons then collide with more atomic nuclei, splitting more atoms and releasing yet more energy in a nuclear chain reaction that increases exponentially.
At about 8:15 in the morning of August 6 1945, the atomic bomb known as the Little Boy detonated releasing energy equivalent to 30 million sticks of dynamite in an instant, devastating the Japanese city of Hiroshima. The surrender of Imperial Japan was announced nine days later. This is the awesome power of exponential growth.
What exponential growth isn’t
Although the concept of exponential growth is not new in the public consciousness, a lot of misconceptions surround the idea. Exponential is often used as a byword for rapid or large.
As a counterpoint, consider the money in your bank account.
Provided the interest is compounded (that is, interest is added to your initial amount and earns interest itself) then the total amount of money in your account increases in proportion to its current size – the hallmark of exponential growth.
As Benjamin Franklin put it: “Money makes money, and the money that money makes, makes more money.” If you could wait long enough, even the smallest investment would become a fortune.
But don’t lock up your rainyday fund just yet. If you invested £100 at 1% per year it would take you over 900 years to become a millionaire.
Very few people would accuse the exponential growth associated with their bank account of being large or rapid.
So exponential growth does not necessarily deal with big quantities, and it is not necessarily fast.
Unfortunately, across a wide range of different countries, the term exponential is appropriate to describe the rapid spread of SARSCoV2, the virus that causes COVID19.
With cases doubling every three to four days in the UK and deaths doubling every two to three days, things could get ugly quickly.
Small comfort
One small crumb of comfort is that almost nothing can grow exponentially forever.
The only exception, ironically, is the alltooslow growth of money in your bank account, which, at least on paper, could grow indefinitely.
Unfortunately, COVID19 cases don’t have to grow exponentially forever, or even for much longer, before the disease becomes one of the most devastating pandemics the world has ever seen.
Because the exponential proliferation of the disease is so undeniably dramatic, any changes we can make at this relatively early stage can make a huge difference even a few days down the line. Now is the time to act, to “double down” on our containment efforts to bring the exponential spread under control.
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