- Exponential Functions
- Exponential Sequence
- Base Numbers
- Natural Exponential Function
- Exponential Model Building (TI-89)
- Nth Root Functions
Watch the video for an overview of exponential functions.
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Exponential functions have the variable x in the power position. For example, an exponential equation can be represented by:
f(x) = bx.
Like other algebraic equations, we are still trying to find an unknown value of variable x.
One way to think of exponential functions is to think about exponential growth—the idea of small increases followed by rapidly increasing ones. These increases (or decreases when working with negative exponents) are consistent over a definite period of time as a function of the variable x. For example, the increases are consistently double or triple. The rapid increases characteristic of exponential functions can be seen on the graph below:
Most exponential graphs will have this same arc shape; There are some exceptions. For example, the graph of ex is nearly flat if you only look at the negative x-values:
Exponential functions are an example of continuous functions.
Graphing the Function
The base number in an exponential function will always be a positive number other than 1. The first step will always be to evaluate an exponential function. In other words, insert the equation’s given values for variable x and then simplify.
For example, we will take our exponential function from above, f(x) = bx, and use it to find table values for f(x) = 3x.
Step One: Create a table for x and f(x)
Step Two: Choose values for x.
Step Three: Evaluate the function for each value of x. “Evaluate” means to plug the x-values into the equation and solve.
|-2||3-2 = 1/32 = 1/9|
The points in our chart are then plotted on the x-axis and y-axis of our graph giving us the following:
Real World Uses
Exponential functions can also be applied in real world contexts to determine things like population growth and radioactive decay. In these cases, the function will not be like above (f(x) = bx) but rather the formula must account for other factors. For example, in the case of radioactive decay, the formula would look like this:
- R = the remaining value of the substance
- A = the initial amount of the substance (grams in the example)
- h = the half-life of the substance
- t = the amount of time passed (60 years in example)
An exponential model has a distinctive upward or downward curve that increases (or decreases) sharply and smoothly. If the curve decreases, it’s called exponential decay; If the curve increases, then it’s exponential growth.
Many real life data sets follow an exponential pattern, including population growth and decline, environmental concentrations (Ott, 1995) and—oddly—even the amount of revenue collected yearly by the IRS (Larson & Falvo, 2012, p. 380).
Mathematically, exponential models have the form y = A(r)x, where A is the initial value, and r is the rate of increase (or decrease). The image above shows an exponential function N(t) with respect to time, t. The initial value is 5 and the rate of increase is et
Exponential growth is an increase in some quantity that follows the relationship
N(t) = A e(kt)
where A and k are positive real-valued constants.
Before diving further into the mathematics, let’s look at a graph of exponential growth. This plot assumes that A = 3 and k = 1.
The function’s initial value at t = 0 is A = 3. The variable k is the growth constant. The larger the value of k, the faster the growth will occur.
The exponential behavior explored above is the solution to the differential equation below:
dN / dt = kN
The differential equation states that exponential change in a population is directly proportional to its size. Initially, the small population (3 in the above graph) is growing at a relatively slow rate. However, as the population grows, the growth rate increases rapidly.
Exponential Growth: Example Problems
Exponential growth can be found in a range of natural phenomena, from the growth of bacterial populations to the speed of computer processors.
Problem 1: A colony of bacteria doubles its population every 4 hours. If the colony originally has ten bacteria, how large will the colony be 24 hours later?
Solution: Since the colony has an original population of 10, then A=10. Knowing that the population will be 20 four hours later, we can solve for the growth constant.
- N(t) = A e(kt)
- 20 = 10 e(k * 4 hours)
- ln(2) = (4 hours) * k
- k = 0.173 /hours
Then, the growth constant can be used to determine the population’s size one day later.
- N(24 hours) = 10 e(0.173 /hours * 24 hours)
- N(24 hours) = 635
Amazingly, the original handful of bacteria will blossom into a colony of nearly a thousand in one day’s time. That’s the power of exponential growth.
Problem 2: A client deposits $100 in a savings account at the local bank. The account grows by 1% interest, compounded annually. What will the value of the account be after ten years?
Solution: The initial size of the account is $100, so A=100. The account’s value will be $101 after one year, due to the interest. Knowing this, we can calculate the growth constant.
N(t) = A e(kt)
101 = 100 e(k * 1 year)
ln(1.01) = (1 year) * k
k = 0.00995 /years
To find the value of the account at ten years, t = 10.
N(10 years) = $100 e(0.00995/years * 10 years)
N(10 years) = $110.46
Exponential decay is a decrease in a quantity that follows the mathematical relationship. In many ways you can think of it as the opposite of exponential growth: where exponential growth goes up, exponential decay goes down.
N(t) = A e(-kt)
where A and k are positive, real-valued constants.
The following graph shows exponential decay, where A = 5 and k = 1.
The function’s initial value at t = 0 is A = 5. k is a variable that represents the decay constant. The larger the value of k, the faster the decay will happen.
Half-Life in Exponential Decay
The half-life is the time after which half of the original population has decayed. From the language of our original exponential decay equation, the half-life is the time at which the population’s size is A/2. Then, by plugging this value into our equation, we arrive at an expression for the half-life:
- A/2 = A e(-kt)
- ½ = e(-kt)
- ln(0.5) = -kt
- t = -ln(0.5)/k
Problem 1: The half-life of carbon-14 is 5,730 years. What is its decay constant?
Solution: To solve, we can use the equation for half-life.
- t = -ln(0.5)/k
- 5730 = 0.693/k
- k = 1.21 * 10(-4) /years
The exponential behavior explored above is the solution to the differential equation:
dN/dt = -kN
The differential equation states that exponential change in a population is directly proportional to its size. For a large population, the decay is rapid. In contrast, as the population shrinks in size, the rate of decay becomes slower.
Radioactivity is the most common natural example of exponential decay. Over time, an unstable atom will eject particles from its nucleus. As these particles discharge, less radioactive material remains. Carbon dating, cancer therapies, and x-ray machines all involve radioactivity.
Problem 2: A wooden tool recovered from an archaeological site contains 65% of its original composition of carbon-14. Approximately how long ago was the wooden tool created?
Solution: Because carbon-14 decays due to radioactivity, we can use the exponential decay equation.
N(t) = A e(-kt).
From Problem 1, the decay constant for carbon-14 is 1.21 * 10(-4) /years. Additionally, if the original population of carbon-14 is A, then N(t) = 0.65A.
0.65A = A e(-1.21 * 10(-4) t)
ln(0.65) = -1.21 * 10(-4) t
t = 3560 years ago
Exponential Growth and Decay: References
Matthews, John A. “Exponential Growth.” 2014: 387–387. Print.
“Exponential Decay.” The Penguin Dictionary of Physics. United Kingdom, Penguin, 2000.
An exponential sequence e(n) is a list of numbers that follows the formula
e(n) = An.
Exponential Sequence Example
If A is a real number, then e(n) is called a real sequence. For example, if A is 3, then the first four terms in the sequence are:
- 31 = 3
- 32 = 9
- 33 = 27
- 34 = 81.
Relationship to Geometric Sequences
A geometric sequence is a list of terms, where the next term is obtained by multiplying by the same amount (a common ratio) to get the next term. The above sequence 3n has 3, 9, 27, and 81 as the first four terms, each of which can be obtained by multiplying the term before it by 3:
- 3 * 3 = 9
- 9 * 3 = 27
- 27 * 3 = 81.
Finding the nth Term of an Exponential Sequence
Finding a formula for an exponential sequence is quite involved and there isn’t a formula you can follow to find it. However, knowing that it behaves as a geometric sequence makes it a lot easier to find the nth term of the sequence. For example, consider the following question:
Example question: What is the next term in this exponential sequence?
The easiest way to answer this type of question isn’t to figure out what An is. Instead, you know it behaves like a geometric sequence, so look for the common ratio. A quick glance tells us that the denominator is multiplied by 4 each time (4 * 4 = 16 and 16 * 4 = 64). So the denominator for the next term in the sequence is 64 * 4 = 256, so the term is 1/256.
The term “base number” in calculus usually refers to the number found in exponential functions, which have the form
The “a” is the above expression is the base number, so-called because it’s a solid base(foundation) for the rest of the expression—like the hovering exponent (x) to the right. The base can be any positive real number not equal to 1.
A base number system is, like the name suggests, an entire number system based on a certain number of digits. For example, the decimal number system has 10 digits.
The base number refers to the number that is multiplied by itself in an exponent. It’s usually written in a larger font.
The exponent tells us how many times to multiply the base number by itself. It’s usually written in a smaller font (as a superscript).
In the above image, the base number is 8. In other words, multiply 8 by itself. The exponent in this example is 3. What this tells you is that you should multiply 8 by itself 3 times:
8 x 8 = 64 x 8 = 512
The term “Base number” also refers to the values in a number system. More specifically, the base tells you how many digits are in the system. For example, the base 10 system has 10 digits from 0 to 9.
The number system with the fewest number of digits is the binary system, also known as a base-2 number system. The two digits of the binary system are 0 and 1. Each digit in the binary system is known as a bit. The binary system is commonly used by most computer systems.
The natural exponential function, ex, is the inverse of the natural logarithm ln. The e in the natural exponential function is Euler’s number and is defined so that ln(e) = 1. This number is irrational, but we can approximate it as 2.71828.
It has one very special property: it is the one and only mathematical function that is equal to its own derivative (see: Derivative of ex). Looking at this graphically, this means that the slope of a tangent line to the curve at any point is equal to the height of the curve at that point.
The graph of ex is a reflection of ln(x) over the line y = x.
The limit of ex as x goes to minus infinity is zero, and the limit as x goes to positive infinity is infinity.
These two functions are inverses of each other:
Properties of the Natural Exponential Function
ex has some handy properties that make it very easy to manipulate algebraically. Because of the laws of exponents, we know that:
Why Use Natural Exponential Functions?
We don’t just use the natural exponential function because it makes algebra become easy, though. It’s also an accurate model of many processes in our world, including the growth and decay of biological systems and what happens to the money when it is compounded.
In simple terms, it does the opposite, or “undoes” the exponential. For example, if x = 2, the exponential function 2x would result in 22 = 4. The nth root (in this case, the cube root, √) takes the output (4), and gives the original input: √(4) = 2.
The term “nth” just means an order: 1st, 2nd, 3rd, and so on. It is just shorthand notation for the entire group of endless possibilities.
The nth root function can be written in two ways:
n√· or, equivalently, x1 / n.
The second format—x1 / n— is what you’ll want to put into a graphing program or calculator. Some calculators do have nth root functions, like the Casio FX-9750G (press SHIFT and ).
The graph above shows an nth root function where n = 2. When n = 2, the function is called a square root function. It’s just a specific nth root with a specific name.
Another example of an nth root function is a cube root function, where n = 3:
For example, the cube root of -1 is -1: 3√(-1) = -1.
Formal Definition of an nth Root Function
The formal definition is:
n√· : [0, ∞] ℝ, given by
n√ (x) = the unique real number y ≥ 0 with yn = x.
The nth root function, n√(x) is defined for any positive integer n. However, there is an exception: if you’re working with imaginary numbers, you can use negative values. For example, (-1)½ = ± i, where i is an imaginary number.
Properties of the nth root Function
The nth root function is a continuous function if n is odd. If n is even, the function is continuous for every number ≥ 0. Note though, that if n is even and x is negative, then the result is a complex number.
Exponential Model Building on a Graphing Calculator
Some graphing calculators (most notably, the TI-89) have an exponential regression features, which allows you to take a set of data and see whether any exponential functions would be a good fit.
Exponential regression fits an exponential function to your data. As an example, let’s say you have the following data:
- x-values: 1, 2, 3, 4, 5, 6, 7,
- y-values: 334, 269, 193, 140, 105, 67.
You might notice that the data decreases sharply, so a decreasing exponential function might be a good fit.
Step 1: Make a scatter plot. Watch the first minute of this video if you don’t know how to create one. This step confirms that the data roughly fits an exponential model. If your data doesn’t fit the model, stop here. You could (theoretically) continue, but your model will be practically useless. Find another model that better fits your data.
Step 2: Press APPS, then scroll to Data/Matrix Editor(using the cursor keys). Press ENTER.
Step 3: Press 1 “Current”.
Step 4: Press F5 “Calc”. A new screen will open.
Step 5: Move the cursor to “Calculation Type”, then press the right-cursor key and choose “4:ExpReg”.
Step 6: Enter your x-values location into the “x” box. For example, if your x-values are in list a1 then type “a1.”
Step 7: Enter the location of your y-values into the “y” box.
Step 8: Move the cursor to the Store ReqEQ line. Press the right cursor key, then move the cursor to y1(x). Press ENTER.
That’s it! A window will appear with a and b . These go into the regression equation y = abx. The same equation will also show in the y1= line of the Y= screen.
If you entered the data in the above example, you should get a solution of y = 490.631792*.726657x.
Tip: Y-values must be greater than zero in order for regression to work prroperly.
Lost your guidebook? You can download a new one from the TI website. The Titanium Guidebook can be found on this page.
Exponential Function: References
Chapter 1 Review: Supplemental Instruction. Retrieved December 5, 2019 from: https://apps-dso.sws.iastate.edu/si/documentdb/spring_2012/MATH_165_Johnston_shawnkim_Chapter_1_Review_Sheet.pdf
Ellis, R. & Gulick, D. (1986). Calculus with Analytic Geometry. Harcourt Brace Jovanovich
Math 142a Winter 2014. Retrieved December 5, 2019 from: http://www.math.ucsd.edu/~drogalsk/142a-w14/142a-win14.html
Larson, R. & Falvo, D. (2012). Algebra and Trigonometry: Real Mathematics, Real People. Cengage Learning.
Nau, R. The Logarithmic Transformation. Retrieved February 24, 2018 from: https://people.duke.edu/~rnau/411log.htm
Ott, W. Environmental Statistics and Data Analysis, Taylor & Francis. 1995.
Pike, S. (2021). Geometric Sequences. Retrieved January 20, 2021 from: http://www.mesacc.edu/~scotz47781/mat150/notes/sequences/Geometric_Sequences.pdfVing, Pheng Kim. Calculus of One Real Variable. Chapter 7: The Exponential and Logarithmic Functions. Retrieved from http://www.phengkimving.com/calc_of_one_real_var/07_the_exp_and_log_func/07_01_the_nat_exp_func.htm on July 31, 2019
Woodard, Mark. 7.3 The Natural Exp. Lecture Notes. Retrieved from http://math.furman.edu/~mwoodard/math151/docs/sec_7_3.pdf on July 31, 2019
Pilkington, Annette. Calculus 2 Lecture Slides. Lecture 3. Retrieved from https://www3.nd.edu/~apilking/Calculus2Resources/Lecture%203/Lecture_3_Slides.pdf