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K12MATH012: Precalculus I

Unit 4: Exponential and Logarithmic Functions   Our final unit concludes the course with the exploration of exponential and logarithmic functions. These two functions are studied together because they create inverses of each other.
 
Exponential functions describe situations that get very large or very small quickly. The unknown in the function is in the exponent. Solution processes for solving these exponential equations are significantly different from how we handle equations involving linear, polynomial, or rational functions. Of course, equations like 3x = 9 or 2x = 16 can be solved easily using number sense, but equations like 7x = 10 need special techniques, and that’s where logarithms come in.
 
Logarithms were invented in the early 17th century as a way to simplify some calculations. For example, if you need to multiply 7,000 by 80,000, you can multiply 7 * 8 = 56 and add seven zeros thereafter, yielding 560,000,000. Technically, you are doing the follow calculation:
 
7,000 * 80,000 = (7 * 103) (8 * 104) = (7 * 8) (103 * 104) = 56 * 107 = 560,000,000.
 
Logarithms use properties of exponents to simplify calculations. One of the advantages of logarithms is that problems like 7x = 10 can be written in a different format, and that simplifies the solution process. For a long time, those logarithmic expressions were approached by books of tables, but we live in the world of technology, where a simple scientific calculator can yield the information we want.

Unit 4 Time Advisory
Completing this unit should take approximately 23 hours and 50 minutes.

☐    Subunit 4.1: 3 hours and 35 minutes

        ☐    Subunit 4.1.1: 40 minutes

        ☐    Subunit 4.1.2: 1 hour and 5 minutes

        ☐    Subunit 4.1.3: 1 hour and 50 minutes

☐    Subunit 4.2: 2 hours and 20 minutes

        ☐    Subunit 4.2.1: 30 minutes

        ☐    Subunit 4.2.2: 1 hour and 50 minutes

☐    Subunit 4.3: 4 hours and 35 minutes

        ☐    Subunit 4.3.1: 45 minutes

        ☐    Subunit 4.3.2: 45 minutes

        ☐    Subunit 4.3.3: 50 minutes

        ☐    Subunit 4.3.4: 2 hours and 15 minutes

☐    Subunit 4.4: 4 hours and 15 minutes

        ☐    Subunit 4.4.1: 1 hour and 15 minutes

        ☐    Subunit 4.4.2: 30 minutes

        ☐    Subunit 4.4.3: 2 hours and 30 minutes

☐    Subunit 4.5: 2 hours and 30 minutes

        ☐    Subunit 4.5.1: 45 minutes

        ☐    Subunit 4.5.2: 1 hour and 45 minutes

☐    Subunit 4.6: 3 hours and 55 minutes

        ☐    Subunit 4.6.1: 55 minutes

        ☐    Subunit 4.6.2: 50 minutes

        ☐    Subunit 4.6.3: 2 hours and 10 minutes

☐    Subunit 4.7: 2 hours and 40 minutes

        ☐    Subunit 4.7.1: 40 minutes

        ☐    Subunit 4.7.2: 2 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Solve problems involving exponential growth and decay.
  - Solve problems involving compound interest and annual percentage yield.
  - Apply continuous growth models in solving problems.
  - Create graphs of exponential functions.
  - Evaluate exponential functions and apply transformation techniques to create graphs of more complex exponential functions.
  - Translate between exponential and logarithmic forms of representation.
  - Translate a logarithm to a different base.
  - Solve exponential equations.
  - Use properties of logarithms in solving logarithmic equations.
  - Graph logarithmic functions.
  - Evaluate logarithmic functions and apply transformation techniques to create graphs of more complex exponential functions.
  - Evaluate real-life problems and apply techniques of logarithmic and exponential functions in order to reach a solution.
  - Fit exponential equations to data.

Standards Addressed (Common Core):
- CCSS.Math.Content.HSA-SSE.A.1 - CCSS.Math.Content.HSA-SSE.A.2 - CCSS.Math.Content.HSA-SSE.B.3 - CCSS.Math.Content.HSA-CED.A.4 - CCSS.Math.Content.HSA-REI.D.10 - CCSS.Math.Content.HSA-REI.D.11 - CCSS.Math.Content.HSF-IF.A.2 - CCSS.Math.Content.HSF-IF.B.4 - CCSS.Math.Content.HSF-IF.B.5 - CCSS.Math.Content.HSF-IF.B.6 - CCSS.Math.Content.HSF-IF.C.7 - CCSS.Math.Content.HSF-IF.C.8 - CCSS.Math.Content.HSF-BF.B.3 - CCSS.Math.Content.HSF-BF.B.4 - CCSS.Math.Content.HSF-BF.B.5 - CCSS.Math.Content.HSF-LE.A.1 - CCSS.Math.Content.HSF-LE.A.2 - CCSS.Math.Content.HSF-LE.A.3 - CCSS.Math.Content.HSF-LE.A.4 - CCSS.Math.Content.HSF-LE.B.5 - CCSS.ELA.Literacy.RST.11-12.1 - CCSS.ELA.Literacy.RST.11-12.2 - CCSS.ELA-Literacy.RST.11-12.3 - CCSS.ELA-Literacy.RST.11-12.4 - CCSS.ELA.Literacy.RST.11-12.5 - CCSS.ELA-Literacy.RST.11-12.10

4.1 Exponential Functions   Exponential functions have their unknown in the exponent of the function. Suppose you invest $2 in a bank that doubles your money every month. The amount of money you have after x months could be described by the exponential function f(x) = 2x. Note that after a year, you would have f(12) = 212 = $4,096. That’s fast money! Some exponential functions get small very fast. This subunit will help you identify exponential functions, to find equations to describe them and to apply them to the concept of compound interest. 

4.1.1 Defining Exponential Functions   You are headed into complex territory. Pay close attention to the definition of an exponential function, and make sure you understand each of the pieces. This subunit begins our exploration and sets the stage for understanding the exponential counterpart you will encounter later - the logarithmic function.

4.1.2 Finding Equations of Exponential Functions   The number of chain letter recipients, the growth of a population, the decay of a hot dog over time - each of these situations can be described by an exponential function. This subunit focuses on extending the ideas of exponential functions and how to find equations to describe them.

4.1.3 Compound Interest   If you get compound interest, that means you get interest on any interest you earned before - it compounds. If you deposited $1,000 at 10% annual interest, at the end of one year, you would have $1,100, and at the end of the second year, another $100 dollars in interest would yield $1,200. However, if you have compound interest, at the end of the second year, the bank account would hold $1,320. This subunit addresses an important application of exponential functions - compound interest.

4.2 Graphs of Exponential Functions   This subunit looks at the characteristics of exponential function graphs. Look for asymptotes, intercepts, intervals of increase, and intervals of decrease. After identifying the customary features of the graphs, you will have an opportunity to apply transformations to exponential functions.

4.2.1 Features of Graphs of Exponential Functions   Starting with the big picture of a common exponential function, this portion of the course helps you develop an internal picture of the behavior of these functions. Pay attention not only to the big picture in the graphs but also to the specific details, as doing so will be helpful in the application of transformations in the next part of Subunit 4.2.

4.2.2 Transformations of Exponential Graphs   Graphing is easier if you develop an internal framework of basic functions and then use transformation concepts to be able to envision more complex functions. This subunit applies transformations to graphs of exponential functions. You’ll find the same concepts you studied before and now apply them to these more complex functions. 

4.3 Logarithmic Functions   Logarithms are an alternative way to write information that is written in exponential form - much like y = x + 2 is the same information as x = y - 2. Logarithms make it possible to solve complicated exponential equations like the 7x = 10 identified in the first sentences of Unit 4. The inverse of an exponential function is a logarithm.

4.3.1 Logarithms and Exponentials   This subunit demonstrates the connections between logarithmic and exponential form. Begin by thinking in terms of translation; that perspective will be a way to simplify your thinking later when the relations become particularly complex.
 
The following two symbolic forms yield exactly the same information.

23 = 8 and log28 = 3

At first students often see no point in a new (and confusing!) way of writing information. The logarithmic form, however, simplifies working with exponential functions. This form gives us a relatively easy way to calculate population growth, compound interest, and speed of decay among other applications. - Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions (PDF)
 
Instructions: Read pages 242 - 244. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 243. Answers are on page 250.
 
Completing this activity should take approximately 35 minutes.
 
Standards Addressed (Common Core):

-   [CCSS.Math.Content.HSF-IF.C.8](http://www.corestandards.org/Math/Content/HSF/IF/C/8)
-   [CCSS.Math.Content.HSF-BF.B.4](http://www.corestandards.org/Math/Content/HSF/BF/B/4)
-   [CCSS.Math.Content.HSF-BF.B.5](http://www.corestandards.org/Math/Content/HSF/BF/B/5)
-   [CCSS.Math.Content.HSF-LE.A.4](http://www.corestandards.org/Math/Content/HSF/LE/A/4)
-   [CCSS.ELA-Literacy.RST.11-12.4](http://www.corestandards.org/ELA-Literacy/RST/11-12/4)
-   [CCSS.ELA-Literacy.RST.11-12.10](http://www.corestandards.org/ELA-Literacy/RST/11-12/10)

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  • Did I Get This? Activity: mathcentre: “Logarithms: Diagnostics” Link: mathcentre: “Logarithms: Diagnostics” (Flash)
     
    Instructions: Click on the link above to check your logarithms skills. A check when you submit an answer indicates your answer is correct; an x when you submit an answer indicates your answer is incorrect. Work until all answers are correct.
     
    Completing this activity should take approximately 10 minutes.
     
    Standards Addressed (Common Core):

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4.3.2 Common and Natural Logarithms   Although there are an infinite number of choices for the base of a logarithm, standard practice is to use either common (base 10) or natural (base e) logarithms. Like the number π, the number e (sometimes called “Euler’s number) is irrational and arises from real-world phenomena. Although confusing at first, it is a very convenient number to use as a base for a logarithm.

4.3.3 Two Logarithmic Properties   Remember that a logarithm is an exponent. This subunit begins by applying an important exponent rule to logarithms, and it makes working with them much easier. The material then develops another property - a simple way to change the base of a logarithmic expression.
 
The two common bases for logarithms are easy to access with technology, but often information comes in another base. The second logarithmic property you will learn about in this subunit is how to change from one base to another. Thus if a logarithmic expression is in base 2, you will be able to change the expression to a common or natural logarithm, which can be solved with a few taps of your calculator.

4.3.4 Solving Exponential Equations   Exponential functions and logarithmic functions are usually taught together because of their inverse relations. We solve exponential equations by using logarithms. Facility with translation between exponential and logarithmic forms makes the solution process much easier.

4.4 Logarithmic Properties   Because logarithms are exponents, there are properties that allow us to use addition instead of multiplication, and subtraction instead of division, in certain circumstances. This subunit begins by identifying some important properties, then continues with a historic perspective on logarithms, and concludes by applying logarithmic properties to solving equations.

4.4.1 Exploring Properties of Logarithms   Logarithms are confusing at first, and much of it is the unfamiliar notation. Be patient and keep working with them - they really do get easier over time. Sometimes it helps to translate back and forth between exponential notation and logarithmic notation. This subunit explains properties and connects previous ideas about exponents to the logarithmic properties.

4.4.2 Logarithm Tables and History   Imagine needing to know the solution to 2x = 15 so you could build your perfect castle and having to do the work without a calculator. It’s clear that x is a little smaller than four, but then what? Leonhard Euler (pronounced “Oiler”) developed a complex process involving changing bases and a certain amount of trial and error and good old number sense. Because any process for finding a logarithm by hand was so complex, tables of logarithms were published so the amounts could be looked up. Lucky for us, there are calculators that allow pushing a button to determine a logarithm rather than checking those long lists of tables (and hoping that whoever developed the table didn’t make a careless error along the way). 

4.4.3 Logarithmic Properties in Solving Equations   An exponential equation is often solved by translating it into logarithmic form and using logarithmic properties to solve it more easily. This subunit demonstrates how to apply those logarithmic properties to solve equations, setting you up to be able to solve application problems later in Unit 4.

4.5 Graphs of Logarithmic Functions   The inverse of a logarithmic function is an exponential function. The relationship between the two is more clearly demonstrated by looking at their graphs. This subunit will explore the graphs of logarithmic functions and then apply transformations to connect graphs to more complex functions.

4.5.1 Graphical Features of Logarithmic Functions   Remember that inverse functions are reflections over the line y = x. If you can graph an exponential function, you can graph its logarithmic inverse. The graphs should help you see the relationship between the two types of functions, and each can help you understand the other.

4.5.2 Transformations of the Logarithmic Function   Now that you are able to graph simple logarithmic functions, you can explore how to apply concepts of transformations in order to graph more complex functions. The fact that the inverse of a logarithmic function is an inverse function can help give you insights into what to expect for results.

4.6 Exponential and Logarithmic Models   Many real-life situations are modeled with exponential or logarithmic functions - the speed at which an investment will double, the length of time radioactive material would take to decay to a negligible amount, the time of death based on the temperature of a body and the strength of earthquakes, for example. This subunit addresses these functions and provides you with opportunities to utilize these models in application problems.

4.6.1 Radioactive Decay and Doubling Time   How long does it take for radioactive waste to decay? How fast is the growth of a population? This subunit develops functions to describe growth and decay and offers application problems to explain some uses of exponential functions.

4.6.2 Newton’s Law of Cooling   It’s cold and rainy outside and you just spent a long, intense time finding inverses of exponential functions. You find tempting eggrolls in the freezer and pop them in the microwave. As you open the door, the tasty smell wafts your way, and you can hardly wait. BUT . . . The eggrolls are too hot to eat! You wonder, “How long will it take for them to cool down?” This is a job for Newton’s law of cooling, and you are lucky enough to be ready to investigate this topic in this subunit.
 
Newton’s law of cooling tells us that the rate of change of temperature (of your flavorsome eggroll) is proportional to the difference between the temperature of the eggroll and the temperature of the room.

4.6.3 Logarithmic Scales and Magnitude   Frequently, charts in newspapers and magazines have linear scales, meaning the chart’s scales increase in an additive way, such as of 5, 10, 15, 20, 25 . . . . A logarithmic scale shows a multiplicative difference in between, like 10, 100, 1,000, 10,000 . . . . This subunit discusses logarithmic scales, which are often employed when numbers are very far apart. A logarithmic scale frequently uses the exponents for the scale marks, so 1, 2, 3, 4 . . . . would represent 101,102, 103, 104 . . . . Such a scale on a chart is called a logarithmic scale. 

4.7 Fitting Exponentials to Data   This subunit continues the work of Subunit 4.6.3 by addressing graphs using logarithmic scales. It shows an example of the difference between graphs with linear scales and graphs with logarithmic scales. The subunit concludes by teaching you how to develop an exponential function to a set of data points.

4.7.1 Semi-log and log-log Graphs   When numbers for a graph cover a large range, it is sometimes helpful to graph using logarithmic scales. This subunit focuses on graphs using logarithmic scales and sometimes using both logarithmic and linear scales.

4.7.2 Fitting Exponential Functions to Data   As you have seen, fitting a function to data is a complex process. This subunit demonstrates a way to use linear techniques to develop a function fitting exponential data.

Final Exam   - Final Exam: The Saylor Foundation’s “K12 Precalculus Final Exam” Link: The Saylor Foundation’s “K12 Precalculus Final Exam” (HTML)

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