Relationships With Variable Exponents

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Objectives
  • Introduce the concept of having the unknown variable in an equation be the exponent
  • See and understand the general patterns associated with relationships that involve a variable as the exponent, using x-y tables
  • Further justify evaluating expressions when the exponent is an irrational number
  • Define the concepts of exponential growth and exponential decay, and know how to tell which one describes a given relationship
  • Investigate basic growth and decay models that use rate of growth or decay $r$ and initial quantity $a$
  • Be able to set up and interpret a simple growth or decay model in the context of a word problem
Lesson Description

For the first time, we will begin to understand relationships that involve constants raised to variable exponents instead of the other way around (e.g. $3^x$ instead of $x^3$). We will examine the behavior of these relationships using integer values for $x$ before seeing what happens when we allow exponent variable $x$ to be non-integer values. Finally, we will look at exponential growth versus exponential decay, and begin to work with basic growth and decay models.

 
Practice Problems

Practice problems and worksheet coming soon!

 

Variable Exponents

In the last lesson » we started warming up to the idea that exponents can be any real number, not just integers and fractions. The main reason to start considering the possibility that any real number $x$ could be an exponent is to begin to study functions where it is the exponent that is the variable, not the base.For example, most of our Algebra study to this point has involved variables as the number that we "do stuff to", like raise it to the second power or add 5 to it ($x^2$, $x+5$). Now we're going to let $x$ be the unknown exponent action that we do to another number, such as $2^x$.We often refer to such functions and relationships as exponential.
Define: Exponential RelationshipsAn exponential relationship is one that contains a variable exponent. The number that we raise to the variable exponent power is often called the base.
For example, in the exponential relationship$$y = 2^{x}$$ would call $2$ the base.

Exponential Patterns

Because we're so familiar with polynomials up to this point, many courses and teachers tend to draw a lot of comparisons between polynomial functions and exponential functions, in terms of graph shape and behavior.We can use fairly simple ideas and concepts we already know to understand exponentials. For example, we know that at their most basic level, exponents are repeated multiplication. This leads to some intuitive rules about the base number.
  • If the base number is larger than $1$, then the value of an exponential will increase as it is raised to larger and larger powers
  • If the base number is a positive fraction smaller than $1$, then the value of an exponential will decrease as it is raised to larger and larger powers
  • If the base number is $1$, then all values of the exponential would be $1$ because $1$ raised to any real number exponent is equal to $1$
For this reason, we will not study exponential relationships or functions that have a base of $1$. They are moot relationships because the variable doesn't matter, since $1^x$ is $1$ for any value of $x$.We will also not study relationships with negative base numbers. Those types of relationships are not continuous lines when graphed because their results flip back and forth between positive and negative, infinitely quickly (we would need a little Calculus to prove this but just know that you won't see this type of relationship in any algebra or Calculus course).Without any further discussion, let's look at the graphs of two exponential functions, one from each case (one with a fraction base and one with an integer base): $f(x) = 2^{x}$ and $g(x) = (1/2)^{x}$As described, we're seeing that as $x$ gets larger, so does the result of the function $f(x)$ (since we are raising a number like $2$ to bigger and bigger powers). Also as expected, the graph of $g(x)$ is decreasing as we raise it to larger and larger powers. This is because repeated multiplication of a fraction with itself decreases the value of the result. Perhaps you are familiar with this from your number knowledge up to this point - for example if you take your calculator and multiply any number smaller than $1$ by itself over and over (even something close to $1$, like $0.99$) you'll see that the result continues to shrink.Eventually it approaches $0$, which we'll continue to talk about as we learn more about these relationships.

Growth and Decay

As we said, the two major cases of exponential behavior we described above are based on whether the base number is a fraction or a number larger than $1$. We are expected to be familiar with the properties of each case, including their names.
Define: Exponential Growth and DecayWhen the base number is greater than $1$, an exponential relationship of the form$$y = b^{x}$$is called exponential growth.When the base number is a fraction between $0$ and $1$ (not inclusive), an exponential relationship of the form$$y = b^{x}$$is called exponential decay.
We will study graphs of exponential functions in an upcoming lesson » but it is helpful to start remembering what shape the graph of each case generally has (the two graphs above). In these non-linear graphs, the "flat" side slope gets less steep and gets closer to zero the further you go out, and the "steep" side grows upward very, very quickly. The slope direction (positive versus negative) never switches. These properties set exponential functions apart from polynomial ones.

A Closer Look

It is worth understanding how these patterns work by recalling how each case will evaluate with both positive and negative exponents.
 
Example 1Fill in the following $x$-$y$ chart for the relationship $y=2^x$, and plot the points in the coordinate plane.$$ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \\ \hline -2 & \\ \hline -1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline \end{array} $$$\blacktriangleright$ Using the properties of exponents, we should be able to generate the following complete table.$$ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \frac{1}{8} \\ \hline -2 & \frac{1}{4} \\ \hline -1 & \frac{1}{2} \\ \hline 0 & 1 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline \end{array} $$Now, plotting these coordinates is a routine task:It is clear that we are looking at a similar graph to what we were told to expect for an exponential growth relationship. By looking at the positive and negative numbers we used to make this plot, we can better understand why the left-side behavior approaches zero and why the right side behavior grows in value very quickly based on what it means to raise $2$ to a large negative number versus a large positive number.Now let's look at an $x$-$y$ table for an exponential decay relationship.
 
Example 2Fill in the following $x$-$y$ chart for the relationship$$y=\left(\frac{1}{3}\right)^x$$and plot the points in the coordinate plane.$\blacktriangleright$ Let's follow the same approach we used for the growth example.$$ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & \\ \hline -1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} $$Fill this in by plugging in each value of $x$ and using what you know about exponents.$$ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & 9 \\ \hline -1 & 3 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{3} \\ \hline 2 & \frac{1}{9} \\ \hline \end{array} $$Plotting these points and graphing the relationship:The shape and pattern we expect for exponential decay is present.

Modelling Growth and Decay

Exponential growth and decay patterns are among the most relatable real-world relationships, and because of that, we see a lot of word problems. Fortunately, they are all very similar in setup and execution.Any phenomenon that grows or shrinks multiplicatively can be described by what I refer to as the "amount function."
Define: The Amount FunctionLet $A_0$ (pronounced "A not" or "A zero") be an initial quantity. If the amount of this quantity changes multiplicatively at a rate of $r$ per unit time $t$, then the quantity present after time $t$ is $A_t$, given by$$A_t = A_0 (1+r)^t$$If $r$ is positive, the amount will grow over time. If $r$ is negative, the amount will shrink over time.
Let's look at some examples, including some hopefully familiar real-world situations.
 
Example 3A $\$300$ stock increases in value by $1\%$ every month. Write a variable expression that represents the value of the stock after $t$ months, and then use your expression to find the value of the stock after two years.$\blacktriangleright$ In this scenario, we start out with $\$300$, so that is our initial amount $A_0$. The growth rate is $1\%$, but we should always work with decimals when performing calculations since that's how we'll want to communicate them to our calculators ($0.01$ in this case). Plugging in, our variable expression for the amount of stock value $A_t$ that we have after $t$ months is$$A_t = 300 \cdot (1.01)^t$$Finally, the answer to the follow-up question about how much money we have after two years is$$A_24 = 300(1.01)^{24} \approx \$380.92$$We must grow the value $24$ times because the $1\%$ growth happens monthly, and there are $24$ months in two years.
 
Remember!
Make sure your growth rate matches your time measurement. If you are given a monthly growth rate, then $t$ should be measured in months, even if you're asked to answer a question about years.
You Should Know
It is common to use notation like I am using in this example, where the subscript describes the number of time periods that have passed since the initial observation. This practice helps keep answer straight because it is common to use growth functions to answer multiple questions. For example, if you were the one who owned that stock, you may want to know what its value will be one, three, and five years from now to understand the pattern.
 
Example 4You take $500$ milligrams of medicine for a headache. The amount of medicine in your blood decreases by $10\%$ every hour. Write a variable expression for the amount of medicine in your blood after $h$ hours, and then determine the amount of medicine in your blood after four hours.$\blacktriangleright$ This is similar to the last problem in every way except that this situation represents exponential decay instead of exponential growth. We know this is the case because the amount of medicine in the blood is decreasing over time.Going back to the definition of the amount function, we see that we'll set up the same structure but that when the amount is decreasing over time, we simply make the growth rate $r$ a negative number.The initial amount is $500$ and the growth rate is $-10\%$ or $-0.10$. Setting up the expression:$$A_h = 500(1 + (-0.10))^h$$$$\longrightarrow A_h = 500(0.90)^h$$That's the general amount function for the amount of medicine in the blood after $t$ hours. Now, to answer the specific question, the amount of medicine in the blood after four hours is$$A_4 = 500(0.90)^4 \approx 328$$There are approximately $328$ mg of medicine in the blood after four hours.
 

Mr. Math Makes It Mean

Growth and Decay in DisguiseIdeally, exponential relationships will be communicated to you with a positive $x$ exponent. Because teachers seek to write tricky test questions, that's not always the case, and it is important that you look at and think about each exponential relationship to make sure you know whether it is growth or decay - because a negative exponent will flip it from one to the other!Consider the relationship$$y = 5^{-x}$$If we're in a rush or our brains are on autopilot, we may classify this relationship as exponential growth because we see the $5$. However:$$5^{-x} = \left( 5^{-1}\right)^x = \left( \frac{1}{5} \right)^x$$which quickly proves that $y=5^{-x}$ is actually an exponential decay relationship.A similar thing happens if you take a fraction base and raise it to a negative exponent:$$y = \left( \frac{4}{7} \right)^{-x}$$This is actually equivalent to$$y = \left( \frac{7}{4} \right)^{x}$$which is a growth function. So the takeaway is, make sure you're given a positive exponent before you classify relationships as growth or decay!Unknown Growth RateWhen working on Amount Function word problems, it's possible to get problems that ask you to solve for the rate of growth rather than the typical problems that ask us to solve for amounts at various times. While you'll soon be a master of solving for anything at all in an exponential relationship using logarithms », you don't need them to solve this question.
 
Example 5The charge left in a certain battery follows an exponential decay pattern. After three months of use, the battery has 6400 mA of output left. After eight months, the battery has only 200mA of output left. Find the monthly battery drain rate.$\blacktriangleright$ The approach is to set up two amount function relationships, one for each piece of given information, and then use exponent properties to solve for $r$.In general, $A_t = A_0 (1+r)^t$, so we know that$$6400 = A_0(1+r)^3$$$$200 = A_0(1+r)^8$$While it is true that we do not know $A_0$, we can divide these two equations to cancel that term:$$\frac{200}{6400} = \frac{A_0(1+r)^8}{A_0(1+r)^3}$$Simplifying,$$\frac{1}{32} = (1+r)^5$$Finally, we can raise both sides to the $(1/5)$ power and solve for $r$.$$\left( \frac{1}{32} \right)^{1/5} = 1 + r$$$$\frac{1}{2} = 1 + r$$$$r = \frac{1}{2} \; \mathrm{or} \; 50\%$$
 

Put It To The Test

 
Example 6Determine whether the following three variable exponent relationships represent exponential growth or decay.$$\begin{array}{ll} \mathrm{a)} & y = 5^x \\ \mathrm{b)} & y = 3^{-x} \\ \mathrm{c)} & y = \left( \frac{3}{4} \right)^{-x} \\ \mathrm{d)} & y = \left( \frac{9}{4} \right)^{x} \\ \mathrm{e)} & y = \left( \frac{5}{8} \right)^{x} \end{array} $$
Show solution
$\blacktriangleright$ Generally speaking, we'll most often be given exponential relationships with positive exponents, like $\mathrm{a)}$, $\mathrm{d)}$, and $\mathrm{e)}$. In those cases, we know the relationship is exponential growth if the base is larger than $1$, and exponential decay if the base is smaller than $1$. Therefore, $\mathrm{a)}$ is growth, $\mathrm{d)}$ is growth, and $\mathrm{e)}$ is decay.For reasons you can re-derive as we did in the lesson above, the rules switch if the variable exponent is negative. Therefore, $\mathrm{b)}$ is decay, and $\mathrm{c)}$ is growth.
 
Example 7Choose at least five $x$ values and use them to fill out an $x$-$y$ table for the relationship $y=4^x$.
Show solution
$\blacktriangleright$ While you can certainly pick any $x$ values you like, hopefully you chose reasonably small ones for your own sanity.Here's a sample solution, although there are multiple correct ways to complete this task.$$ \begin{array}{|c|c|} \hline x & y \\ \hline -1 & 1/4 \\ \hline -1/2 & 1/2 \\ \hline 0 & 1 \\ \hline 1 & 4 \\ \hline 3/2 & 8 \\ \hline 2 & 16 \\ \hline \end{array} $$
 
Example 8A car currently valued at $\$10,000$ loses $5\%$ of its value every year. Write an expression for the value of the car after $x$ years.
Show solution
$\blacktriangleright$ We can model this situation using the Amount Function, letting $A_0$ be $10,000$ and letting $r$ be $-0.05$, since the amount of car value is decreasing over time. Putting it all together:$$A_t = A_0(1+r)^t$$$$\longrightarrow A_t = 10,000 \cdot (0.95)^t$$
 
Example 9The number of cells of a certain bacteria growing in a science lab doubles every 4 hours. If there were $100$ bacteria cells initially, how many bacteria are present after 2 days?
Show solution
$\blacktriangleright$ Once again we have all we need to use the Amount Function setup, with $A_0$ equal to $100$. The only catch is that we need to work carefully to avoid working with the incorrect growth rate or time period.There are two ways to consider the growth rate. First, we could realize that if something is periodically doubling in size, that the exponential base number must be a $2$. Second, the percent change interpretation of doubling in size is to say that it "grew by $100\%$". If you let $r$ be $100\%$ or $1$, the Amount Function yields an exponential of base $2$ since $1+r$ would be $2$ in that case.Finally, we need to use units correctly. There are a few ways to handle the hours to days conversion but for this problem let's let each $4$ hour time interval be one time unit. This means that in $2$ days we will have $12$ such periods ($6$ each day).Plugging in what we know:$$A_t = A_0(1+r)^t$$$$A_{12} = 100(2)^{12} = 409,600$$
 
Example 10Rabbits are artificially introduced to a neighborhood with no predators, and multiply exponentially. After three weeks, there are $5,400$ rabbits. After seven weeks, there are $437,400$ rabbits. Find the weekly growth rate of these rabbits, as well as how many initial rabbits were introduced.
Show solution
$\blacktriangleright$ This is an example of an Amount Function word problem where we are not given the rate of growth. However, because we're given two instances, we will be able to solve for $r$ using $n$-th roots.From the rabbit population at week three we know $A_3$:$$5,400 = A_0(1+r)^3$$and from the info about week seven we know $A_{5}$:$$437,400 = A_0(1+r)^{7}$$Divide the second equation by the first:$$\frac{437,400}{5,400} = \frac{A_0(1+r)^{7}}{A_0(1+r)^3}$$$$\longrightarrow 81 = (1+r)^4$$$$r = 2$$or$$r = 200\%$$Finally, we can find the initial amount by using either of our two original equations:$$5,400 = A_0(3)^3$$$$\longrightarrow A_0 = 200$$
 
Lesson Takeaways
  • Gain further familiarity with relationships that have variable exponents rather than constant exponents
  • Begin to understand observed patterns in exponential relationships
  • Understand the difference between exponential growth and decay and what distinguishes each
  • Learn the standard growth function and know how to use it to solve for unknown quantities

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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