FunctionsExponentials and Logarithms

Overview

In this module you will be learning about two classes of functions called exponentials and logarithms.

Exponential functions have the independent variable (or input) as the exponent. For example, y = 2x . Note that this is type of function is different than power functions where the input is the base, such as y = x2. Many real-world applications use exponential functions ranging from finance to biology.

Logarithmic functions are inverse functions of the exponential, that is, they help us to find values of an unknown exponent, such as by solving 2x = 10. Using a logarithm, we can determine this mysterious input. Further, logarithms have a core set of algebraic properties that can help in simplifying an expression.

Exponential Topics• Exponents and Laws of Exponents• Fractional Exponents• Solving Problems Involving Exponents• Parent Graphs of Exponential Functions• Transformations of Exponential Functions• Exponential Growth and Decay• Applications of Exponential Functions• The Special Number e• Solving Exponential Equations Algebraically• Solving Exponential Equations Graphically

Logarithm Topics• Inverse Functions (review)• Defining the Inverse of an Exponential Function• Finding the Inverse of an Exponential Function• Two Special Logs• History of the Logarithm• Evaluating Logarithmic Expressions• Graphs of Logarithmic Functions• Properties of Logarithms• Curve Fitting• Solving Logarithmic Equations Algebraically• Solving Logarithmic Equations Graphically

Exponents and Laws of Exponents

• Big Ideas: Before studying exponential functions it is important to have grasp of exponents and the laws that govern them. Exponents are nothing more than a way to express repeated multiplication. • xmxn=x(m+n)

• xm/xn=x(m-n)

• (xm)n=xmn

• x0=1• x-n=1/xn

• (xy)n=xnyn and (x/y)n=xn/yn

Exponentials

Exponents and Laws of Exponents

• Virtual Nerd - What are exponents?• http://www.virtualnerd.com/algebra-1/algebra-foundation

s/exponents-definition.php

• Video from Virtual Nerd describing the meaning of exponents.

Exponentials

Exponents and Laws of Exponents

• Khan Academy - Understanding Exponents• Watch these videos in order.• Khan Academy Video: Understanding Exponents• Khan Academy Video: Understanding Exponents 2• Khan Academy Video: Level 1 Exponents• These videos show the meaning between repeated

addition, repeated multiplication and when you should use multiplication rather than an exponent.

Exponentials

Exponents and Laws of Exponents

• Virtual Nerd - Converting numbers to expanded form.• http://www.virtualnerd.com/pre-algebra/factors-fractions-

exponents/exponential-to-expanded-form-conversion.php

• Did you know that exponents are just a quick way to show repeated multiplication? In this tutorial, see how to expand out a value in exponential form to see what it really represents!

Exponentials

Exponents and Laws of Exponents

• Virtual Nerd - Converting numbers from expanded form.

• http://www.virtualnerd.com/pre-algebra/factors-fractions-exponents/powers/exponent-examples/expanded-to-exponential-form-conversion

• Exponential form is a quick way to show that a number should be multiplied by itself a certain number of times. In this tutorial, see how to write a repeated multiplication in exponential form!

Exponentials

Exponents and Laws of Exponents

• Khan Academy - Exponent Rules• https://www.khanacademy.org/math/arithmetic/exponents

-radicals/exponent-properties/e/exponent_rules

• Practice problems from Khan Academy on solving exponent problems by adding, subtracting, or multiplying exponents.

Exponentials

Exponents and Laws of Exponents

• Illustrative Mathematics - Extending the Definitions of Exponents, Variation 1

• http://www.illustrativemathematics.org/illustration_pdfs/395.pdf

• This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t = 0.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Simplifying Expressions Using the Power to a Power Property

• http://braingenie.ck12.org/skills/105540 • Problems look like (25) x (26) =___. These exercises are

simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Simplifying Expressions Using the Power to a Power Property

• http://braingenie.ck12.org/skills/105541 • Problems look like (b7)8 =___. These exercises are

simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Exponents and Laws of Exponents

• BrightStorm - Multiplication and Division Properties of Exponents

• http://youtu.be/kETKV0KJlvw • A video on YouTube from Brightstorm that shows how to

simplify products and quotients involving exponents.• Starting at 1:30 into the video the teacher shows some

easy examples of the properties/laws of exponents using simple numbers that make it easier to understand the rules.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Simplifying Expressions Using the Multiple Exponent Properties Involving Products

• http://braingenie.ck12.org/skills/105545 • Problems look like (73)6 x 78=___. These exercises are

simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Simplifying Expressions Using the Quotient Property of Powers

• http://braingenie.ck12.org/skills/105545 • Problems look like (x/y)7. These exercises are

simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Simplifying Expressions Using Multiple Exponent Properties Involving the Quotient

• http://braingenie.ck12.org/skills/105546 • Problems look like (412/43)2. Make sure you read the

explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Exponents and Laws of Exponents

• Brightstorm - Zero and Negative Exponents• http://youtu.be/pFW3z7EJZso • A video on YouTube from Brightstorm that shows how to

simplify products involving zero and negative exponents.• This video is less than 2 minutes. Again, the teacher

shows the rules with variables, but then does some simple number examples to explain the rules.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Zero Exponents• http://braingenie.ck12.org/skills/105549 • Problems look like 60 =__. Get 5 in a row and move on.

A 31 second video is included if you need additional help.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Negative Exponents• http://braingenie.ck12.org/skills/105550 • Problems look like 5-2 =__. Get 5 in a row and move on.

A 35 second video is included if you need additional help.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Evaluating Exponential Expressions Involving the Zero Exponent Property and Others

• http://braingenie.ck12.org/skills/105551 • Problems look like 20/72 =__. Get 5 in a row and move

on. A 35 second video is included if you need additional help.

Exponentials

Exponents and Laws of Exponents

• BrainGenie - Evaluating Exponential Expressions Involving the Negative Exponent Property and Others

• http://braingenie.ck12.org/skills/105552 • Problems look like 53/5-4 =__. Get 5 in a row and move

on. A 1 minute 27 second video is included if you need additional help.

Exponentials

Exponents and Laws of Exponents• Properties of Exponents• You’ll need a login to access this site (you can create a free login as a

student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome)

• http://www.sascurriculumpathways.com/portal/Launch?id=5061

• “You'll learn how to use the properties of exponents to simplify expressions and solve simple exponential equations. The lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.”

• When you are doing the self-check problems use ^ when you need to enter an exponent. Pay special attention to the “Stop and Think” audio clips. PDFs at the end are especially good.

Exponentials

Exponents and Laws of Exponents

• SAS Curriculum Pathways - Geometric Sequences• You’ll need a login to access this site (you can create a free login as a

student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome)

• http://www.sascurriculumpathways.com/portal/Launch?id=5062

• “You'll learn how to recognize geometric sequences, write a rule for a geometric sequence, and find the nth term in a geometric sequence. The lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.” When you are doing the self-check problems use ^ when you need to enter an exponent. Pay special attention to the “Stop and Think” audio clips.

Exponentials

Fractional Exponents• Big Ideas: Before studying exponential functions it is

also important to understand that exponents don’t have to be integers (...-2,-1,0,1,2…). Exponents can be fractions or decimals. Fractional exponents are another way of representing square roots, cube roots, 4th roots… nth roots. Another name for roots are radicals. These fractional exponents also follow the laws of exponents mentioned: • xmxn=x(m+n)

• xm/xn=x(m-n)

• (xm)n=xmn

• x0=1• x-n=1/xn

• (xy)n=xnyn and (x/y)n=xn/yn

Exponentials

Fractional Exponents• Khan Academy - Basic Fractional Exponents Video• http://youtu.be/lZfXc4nHooo • Pay special attention to the argument that is made

around minute 2:00 and how he extends the idea at minute 3:00. If you don’t understand those then the example around 4:20 into the video should help you understand the concept of fractional exponents.

Exponentials

Fractional Exponents• Khan Academy - Negative Fractional Exponents • http://youtu.be/tn53EdOr6Rw

http://youtu.be/bIFdW0NZ9W4 • The first of these videos takes the law for negative

exponents and the law for fractional exponents and combines them into one rule. It is only 3 minutes long, watch it all.

• The second video is just over 3 minutes. All he does in this video is makes the base of the negative fractional exponent a fraction instead of a whole number.

Exponentials

Fractional Exponents• Khan Academy - Fractional Exponents Practice

Problem Sets• https://www.khanacademy.org/math/arithmetic/exponent

s-radicals/negative-exponents-tutorial/e/exponents_3

• Make sure you can do 5 in a row correctly before moving on.

Exponentials

Fractional Exponents• Khan Academy - Fractional Exponents with

Numerator other than 1• http://youtu.be/S34NM0Po0eA • This video extends fractional exponents beyond (½),

(⅓), (¼) and ( ) and now allows you to simplify ⅕expressions with exponents like (⅔) or ( ).⅗

Exponentials

Fractional Exponents• Khan Academy - Fractional Exponents with

Numerator other than 1 Problem Set• https://www.khanacademy.org/math/arithmetic/exponent

s-radicals/negative-exponents-tutorial/e/exponents_4

• Make sure you can do 5 in a row correctly before moving on.

Exponentials

Fractional Exponents

• Cool Math - Fractional Exponents• http://www.coolmath.com/prealgebra/14-intro-to-radicals

/07-radicals-fractional-exponents-01-87.htm

• This resource color codes the fractional exponent and its base and shows how the problems can be translated to radical expressions in order to simplify them. This resources gives you the basic concept and a few practice problems to try on your own.

Exponentials

Fractional Exponents

• Math Is Fun - Fractional Exponents• http://www.mathsisfun.com/algebra/exponent-fractional.h

tml

• This resource is for the person who likes to read a thorough easy to understand explanation of what fractional exponents are in relation to radicals and see how the rules of exponents still apply. There is also a nice applet on this page that the user can manipulate to see how changing the fractional exponent affects the graph of the function.

Exponentials

Fractional Exponents• Sophia - Fractional Exponents Tutorial• http://www.sophia.org/fractional-exponents-pathway • Nine different teachers explain several topics related to

Fractional Exponents and each topic has a 3 question multiple choice quiz to check for understanding. The topics include: • Numbers raised to a power that is a fraction• The relationship between roots and fractional exponents• Rewriting Fractional Exponents• Converting to Fractional Exponent Form• Multiplication Rule with Fractional Exponents• Division Rule with Fractional Exponents• Power Rule with Fractional Exponents

• Just click “Learn this” to begin

Exponentials

Fractional Exponents

• Wolfram Alpha Can simplify Fractional Expressions• http://www.wolframalpha.com • An online math tool that can give you multiple forms of

the same answer. This is really useful if you need to check to see if your answer is the same as another form. You can type in (⅜)^(½) and it will give you a lot of different representations of the simplification of that number. In some instances it can even show you step-by-step directions on how they came up with those representations.

Exponentials

Solving Problems Involving Exponents

• Big Ideas: All of the exponent rules you know now can be put into context of real-life situations. Quantities that are really small in comparison have negative exponents. Quantities that are large have positive exponents. They can be whole numbers or fractional. Sometimes a problem may require you to compare to of these such numbers together through multiplication or division.

Exponentials

Solving Problems Involving Exponents

• BrainGenie - Simplifying Expressions Using Multiple Exponent Properties Involving Products

• http://braingenie.ck12.org/skills/105553 • Solve at least 5 problems before moving on. If you get

stuck watch the 1 minute 31 second video.

Exponentials

Solving Problems Involving Exponents

• BrainGenie - Solving Word Problems Using Exponent Properties Involving Products

• http://braingenie.ck12.org/skills/105544 • Solve at least 5 problems before moving on. If you get

stuck watch the 1 minute 11 second video.

Exponentials

Solving Problems Involving Exponents

• BrainGenie - Solving Word Problems Using Exponent Properties Involving Quotients

• http://braingenie.ck12.org/skills/105548 • Solve at least 5 problems before moving on. If you get

stuck watch the 1 minute 21 second video.

Exponentials

Parent Graphs of Exponential Functions

• Big Ideas: Before learning how complex exponential functions can be graphed and manipulated to model a real-world situation one must first conquer the key features of the parent function. The parent function of an exponential is in the form f(x)=abx.

Exponentials

Parent Graphs of Exponential Functions

• SAS - Linear, Exponential and Quadratic Models• You’ll need a login to access this site (you can create a free login as a

student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome)

• http://www.sascurriculumpathways.com/portal/Launch?id=6016

• “You'll learn how to compare linear, exponential, and quadratic models; solve a system of linear and exponential equations; and solve a system of linear and quadratic equations. This lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.” You’ll use functions, tables and graphs to understand the difference between these three function families.

Exponentials

Parent Graphs of Exponential Functions

• SAS Curriculum Pathways - Graphing Exponential Functions

• You’ll need a login to access this site (you can create a free login as a student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome)

• http://www.sascurriculumpathways.com/portal/Launch?id=5063

• “You'll learn how to graph an exponential function and identify the key features of an exponential function. This lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.”

Exponentials

Parent Graphs of Exponential Functions

• Cool Math - Graph exponential function of the form f(x)=abx

• http://www.coolmath.com/algebra/Algebra2/06ExponentialsLogs/06_expgraph.htm

• Explanation of how to graph an exponential function with different bases.

Exponentials

Parent Graphs of Exponential Functions

• Virtual Nerd - Graph exponential function of the form f(x)=abx

• http://www.virtualnerd.com/algebra-1/exponents-exponential-functions/growth-definition.php

• “Exponential functions often involve the rate of increase or decrease of something. When it's a rate of increase, you have an exponential growth function! Check out these kinds of exponential functions in this tutorial!”

Exponentials

Parent Graphs of Exponential Functions

• SAS Curriculum Pathways - Graph exponential function of the form f(x)=abx

• Exploring Graphs of Special Functions• http://www.sascurriculumpathways.com/portal/Launch?id

=1436

• “Use the tool's built-in examples (or create your own) to explore families of special functions—absolute value, radical, rational, and exponential. You'll investigate the relationship between the parameters of an equation of a special function and its graph.”

Exponentials

Parent Graphs of Exponential Functions

• SAS Curriculum Pathways - Graph exponential function of the form f(x)=abx

• Describing the Rate of Change of an Exponential Functions

• http://www.sascurriculumpathways.com/portal/Launch?id=95

• “Explore definitions and examples of exponential growth. After reviewing the rate of change of a linear function, investigate the rate of change of an exponential function. You'll summarize what you learn to answer this focus question: How can the rate of change of an exponential function be described?”

Exponentials

Parent Graphs of Exponential Functions

• Purple Math - Graphing Exponential Functions• http://www.purplemath.com/modules/graphexp.htm • Pay special attention to how to shift a graph left, right, up

or down, or how to reflect it over the x-axis or y-axis. The graphs on page 4 are especially useful.

Exponentials

Parent Graphs of Exponential Functions

• BrainGenie - Graphing Exponential Functions by Making a Table

• http://braingenie.ck12.org/skills/106063 • These are practice problems of how to graph an

exponential function by using a table of values. The base can be a whole number or a fraction and the exponent is usually a multiple of x. Include a 1 minute 31 second video if needed.

Exponentials

Transformations of Exponential Functions

• For Dummies - How to Graph and Transform Exponential Functions

• http://www.dummies.com/how-to/content/how-to-graph-and-transform-an-exponential-function.html

• This is only 1 page long and has a good explanation of how a graph looks with a whole number base or a fractional base. It also has a nice summary of which numbers represent vertical and horizontal shifts and which one is a vertical stretch. Read the entire page.

Exponentials

Transformations of Exponential Functions

• BrainGenie - Transforming Exponential Functions• http://braingenie.ck12.org/skills/106064 • This is a set of practice problems that allows you to

recognize a graph of a transformation of the original exponential function without having to make a table of values.

Exponentials

Exponential Growth and Decay

• Virtual Nerd - Definition of Exponential Growth.• http://www.virtualnerd.com/algebra-1/exponents-exponen

tial-functions/growth-definition.php

• Exponential functions often involve the rate of increase or decrease of something. When it's a rate of increase, you have an exponential growth function! Check out these kinds of exponential functions in this tutorial!

• Solve exponential equations of the form y = abx or y = (1 + r)x.

Exponentials

Exponential Growth and Decay

• Virtual Nerd Video: What is Exponential Growth?• http://www.virtualnerd.com/algebra-1/exponents-exponen

tial-functions/growth-definition.php

• A video that is 7 minute 8 second long. “Exponential functions often involve the rate of increase or decrease of something. When it's a rate of increase, you have an exponential growth function! Check out these kinds of exponential functions in this tutorial!”

• Pay attention to how she relates f(x)=abx in comparison to f(t)=a(1+r)t.

Exponentials

Exponential Growth and Decay• Virtual Nerd Video: How Do You Solve a Word

Problem with Exponential Growth?• http://www.virtualnerd.com/algebra-1/exponents-exponen

tial-functions/growth-word-problem.php

• This video tutorial is 6 minutes 42 seconds. “If something increases at a constant rate, you may have exponential growth on your hands. In this tutorial, learn how to turn a word problem into an exponential growth function. Then, solve the function and get the answer!”

• Pay attention to what she says the difference between a linear function and exponential function is.

• There is a nice explanation of what each variable stands for in the Growth Formula f(t)=a(1 + r)t.

Exponentials

Exponential Growth and Decay

• Virtual Nerd - How Do You Solve a Word Problem with Exponential Decay?

• http://www.virtualnerd.com/algebra-1/exponents-exponential-functions/decay-word-problem.php

• A video that is 6 minutes and 12 seconds long. This is similar to the previous video but deals with decay rather than growth. “If something decreases in value at a constant rate, you may have exponential decay on your hands. In this tutorial, learn how to turn a word problem into an exponential decay function. Then, solve the function and get the answer!”

• The Decay Formula is f(t)=a(1 - r)t.

Exponentials

Exponential Growth and Decay

• Virtual Nerd Video: What is the Formula for Compound Interest?

• http://www.virtualnerd.com/pre-algebra/percents/compound-interest-formula-definition.php

• A video that is 6 minutes and 12 seconds long. “Things like bank accounts, loans, investments, and mortgages are a part of life, and almost always, interest is involved. Sometimes, you need to deal with compound interest, so it would be good to know the formula for it! In this tutorial, you'll see the formula for compound interest. Take a look!”

Exponentials

Exponential Growth and Decay• Virtual Nerd -How Do You Use the Formula for

Compound Interest?• http://www.virtualnerd.com/pre-algebra/percents/compou

nd-interest-formula-example.php

• A video that is 4 minutes 52 seconds long. Notice how the Compound Formula A=P(1+r)t looks like f(t)=a(1+r)t, but be careful because t is usually expressed in years, so note what she does with t around minute 3.

• “If you already have a bank account or if you plan to have one in the future, then this tutorial is a must see!”

Exponentials

Exponential Growth and Decay

• BrainGenie - Compounding Interest Annually Problem

• http://braingenie.ck12.org/skills/106059 • These are word problems. There is a 50 second video

linked to help. Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Exponential Growth and Decay

• BrainGenie - Compounding Interest Monthly Problem

• http://braingenie.ck12.org/skills/106060 • These are word problems. There is a 55 second video

linked to help. Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

Applications of Exponential Functions

• SAS Curriculum Pathways - Using Data to Make Good Decisions

• You’ll need a login to access this site (you can create a free login as a student).

• http://www.sascurriculumpathways.com/portal/Launch?id=1052

• Organize, analyze, and use data to make informed decisions. You'll describe patterns and variances in data and explore risks of making predictions without adequate data.

Exponentials

Applications of Exponential Functions

• SAS Curriculum Pathways - The AIDS Epidemic• You’ll need a login to access this site (you can create a free login as a

student), a graphing calculator and graph paper.

• http://www.sascurriculumpathways.com/portal/Launch?id=294

• Examine the number of reported cases of AIDS in the United States during the peak period of the epidemic. You'll calculate the rate(s) at which the disease spread and determine the function(s) best fitting the statistical data according to exponential growth models.

Exponentials

Applications of Exponential Functions

• SAS Curriculum Pathways - Human Population Growth

• http://www.math.colostate.edu/cauchy/Applets/ExponentialGrowth/exponentialgrowth.htm

• Applet to use to discover the effects of changing the rate of growth of the human population.

• Java applet.

Exponentials

Applications of Exponential Functions

• BrainGenie - Radioactive Decay• http://braingenie.ck12.org/skills/106062 • These are word problems. There is a 1 minute 27 second

video linked to help. Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

Exponentials

The Special Number e

• Purple Math - Exponential Functions: The “Natural” Exponential “e”

• http://www.purplemath.com/modules/expofcns5.htm • A nice description on how “e” was discovered and why it

is used.• Read the paragraphs up to where they start talking about

your calculator being able to calculate “e”.

Exponentials

The Special Number e

• Better Explained - An Intuitive Guide to Exponential Functions

• http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

• An explanation of how 2x is a good basis for ex and how ex takes growth continuously instead of doubling or tripling at specific moments.

• This explanation is accompanied by a video, but the explanation below the video does a better job.

Exponentials

Compounding Interest Continually

• BrainGenie - Comnpounding Interest Continually• http://braingenie.ck12.org/skills/106061 • Earlier we looked at compounding interest annually and

monthly. Now we will look at what it would take to compound interest even more often, as often as every moment of time. This will utilize the number e. There is a 1 minute video linked to help.

Exponentials

Solving Exponential Equations Algebraically

• Pauls Online Math Notes - Solving Exponential Algebraically

• http://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx

• There are several samples of how to solve problems that have x in the exponent. This includes problems such as when two bases are the same or can be re-written to be the same number. The second half of this will also explain when you cannot re-write bases to be the same number, which will be the main focus of the second part of this module.

Exponentials

Solving Exponential Equations Algebraically

• Purple Math - Solving Exponential Equations from the Definition

• http://www.purplemath.com/modules/solvexpo.htm • Only pay attention to the 1st page of this website. The

second and third page can be used to explore the second part of this module on logarithms.

• There is a good example on this page that results in two answers because the exponent includes a quadratic equation.

• Pay special attention to the last example.

Exponentials

Solving Exponential Equations Graphically

• YouTube - MathisPower4u• http://www.youtube.com/watch?v=4Wb-T8bcqZM • Good example of why you have to use a graphing

calculator to solve for exponents that are not integers until you finish with this entire module. This example goes over how you can change your table step value or use the intersection functionality of your calculator to find decimal approximations of exponents that are not integers.

Exponentials

Inverse Functions (Review)

Big Ideas: Two functions are said to be inverses if they “undo” one another. We say functions f(x) and g(x) are inverses if:• f(g(x)) = x AND g(f(x)) = x

We denote the inverse of function f(x) as f -1(x), read as “f inverse of x.” Given a graph, the inverse graph can be obtained by reflection about the line y = x since such a transformation interchanges the variables x and y.

Logarithms

Inverse Functions (Review)

• Math is Fun - Inverse Functions• http://www.mathsisfun.com/sets/function-inverse.html• This website includes easy-to-follow descriptions of

inverses along with 3 examples and helpful graphics. Pay special attention to the notation f -1, which stands for the inverse of a function f. There are 8 “your turn” questions at the bottom which will allow you to test your understanding.

Logarithms

Inverse Functions (Review)

• Paul’s Online Notes (Algebra) - Inverse Functions• http://tutorial.math.lamar.edu/Classes/Alg/InverseFuncti

ons.aspx• This website includes three examples of how to find the

inverse of a function, both algebraically and graphically. Different types of functions are included (linear, radical, and rational). Pay special attention to the big blue box in the middle of the page detailing what it means for two functions to be inverses.

Logarithms

Inverse Functions (Review)

• Inverse Functions Quiz (#1)• http://www.math.uri.edu/~bkaskosz/m111s35/mquiz/inv

qu1.html• Use this quick 5 question multiple-choice quiz to test

your understanding of how to use algebra to find the inverse function of a given equation. Click on an answer choice. “TRUE” means you are correct and “FALSE” means incorrect.

Logarithms

Inverse Functions (Review)

• Inverse Functions Quiz (#2)• http://www.mhhe.com/cgi-bin/netquiz_get.pl?qfooter=/u

sr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_2fq.htm&afooter=/usr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_2fa.htm&test=/usr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_2q.txt&answers=/usr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_2a.txt&email=0

• Use this 9 question multiple-choice quiz to test your understanding of inverse functions algebraically, graphically, and in a table.

Logarithms

Defining the Inverse of an Exponential Function

Big Ideas: An exponential function is of the form y = bx for some positive constant b. The inverse is obtained by interchanging x and y, namely x = by . Solving for y, we will use a logarithm and write y = logbx.

Logarithms

Defining the Inverse of an Exponential Function

• Cool Math: What is a Logarithm?• http://www.coolmath.com/algebra/Algebra2/06Exponent

ialsLogs/07_whatsalog.htm• This website includes two step-by-step examples that

show how to define the inverse of an exponential function both algebraically and graphically. Pay special attention to how an exponential function is colored in blue and its inverse (called a logarithm) is colored in green.

Logarithms

Defining the Inverse of an Exponential Function

• Purple Math: Logarithms• http://www.purplemath.com/modules/logs.htm• This website has a lesson introducing the conversion

between logarithmic and exponential form. For now, just pay attention to the first page, especially the animation showing how a log and an exponential function relate. For now, look over pages 1 and 2 of this lesson.

Logarithms

Defining the Inverse of an Exponential Function

• Khan Academy: Converting an Exponential to Logarithmic Statement

• http://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_basics/v/logarithmic-equations-2http://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_basics/v/logarithmic-equations-2

• This 2 minute video shows how to write an exponential statement as a logarithmic one, and vice versa. Pay special attention to how a log is really expressing a power. Also, be sure you master the vocabulary mentioned around 1:36 in the video.

Logarithms

Finding the Inverse of an Exponential Function

Big Ideas: To find the inverse of an exponential function, we have two approaches:1. Algebraically: Switch x and y and solve for y. Ultimately you will need a logarithm as the inverse of an exponential.

2. Graphically: If we graph y = x and reflect the exponential graph about that mirror line, we obtain the inverse graph, which is the graph of a logarithmic function.

Logarithms

Finding the Inverse of an Exponential Function

• YouTube: Finding the Inverse of an Exponential Function

• https://www.youtube.com/watch?v=eZPTwxSN_18• This 3 minute and 38 second video shows how to find

the inverse of a more complex exponential equation, specifically, y = 5*63x-4+7. Pay close attention to how a log used starting at around 1:16 in the video. Also, be sure you understand how at 2:15, a log and an exponential function of the same base are inverses and thus “undo” each other.

Logarithms

Two Special Logs

Big Ideas: In general, we have considered logs of the type logbx where b is the base. There are two very special bases that are often encountered, so important that these are often the only two logs on a calculator.1. Common Log: We write log(x) (note the invisible base) to represent log “base 10”, that is, log10x.

2. Natural Log: We write ln(x), read “natural log of x”, to represent log “base e”, that is, logex.

Logarithms

Two Special Logs

• Purple Math: The Common and Natural Logarithms• http://www.purplemath.com/modules/logs3.htm• This quick module discusses the two special logs with

base 10 and base e, as well as how to evaluate such quantities using a calculator.

Logarithms

Two Special Logs

• Khan Academy: Natural Logarithm with a Calculator

• http://www.khanacademy.org/math/algebra/logarithms-tutorial/natural_logarithm/v/natural-logarithm-with-a-calculator

• This 3 minute and 37 second video shows how to use a graphing calculator to evaluate ln(67). At 2:56, be sure you reflect on if this answer is reasonable.

Logarithms

History of the Logarithm

Big Ideas: So why logs if they are really just a restatement of an exponential? The mathematical development of a logarithm is quite interesting and lead to the ability to quickly calculate more complicated calculations, ultimately, the calculator.

Logarithms

History of the Logarithm

• History of Logarithms• http://www.westcler.org/gh/outtda/pdf_files/History_of_

Logarithms.pdf• This quick PDF discusses how Napier and Briggs

independently invented (or discovered) logarithms. Note that logs originated with astronomical data.

Logarithms

History of the Logarithm

• Logarithms: Brief History and Brief Math• https://www.youtube.com/watch?v=Cc5LhCyd8lM• This 9 minute and 17 second video (enhanced with a

British accent!) presents the state of mathematics in the early 17th century and how logarithms made computations much more manageable.

Logarithms

History of the Logarithm

• YouTube - Napier’s Bones• https://www.youtube.com/watch?v=y8HKGr77EVc• This quick 1 minute and 25 second video showcases an

amazing invention by John Napier that eventually lead to inventions like the slide rule used heavily by engineers in the 1950’s - 1970’s. Can you figure out why this device works in the multiplication problem shown in the video?

Logarithms

Evaluating Logarithmic Expressions

Big Ideas: An “expression” is a mathematical statement that can be evaluated, such as 23 - 5(2). To find the value of a logarithmic expression, we think about what a logarithm means in the context of an exponential. For example, to evaluate the square root of a number, such as 64, we think about that fact slightly differently, namely, as a square. The square root of 64 is 8 since 82 = 64. Similarly, we can calculate the value of log381 by thinking of its meaning as an exponential. If log381 = x, then, 3x = 81, so x = 4. Thus, log381 = 4.

Logarithms

Evaluating Logarithmic Expressions

• The Math Page: Logarithms• http://www.themathpage.com/aprecalc/logarithms.htm#d

efinition• This lesson contains many examples and quick check

questions on how to evaluate simple log expressions. Look through examples 1 through 8 then try problems 2 through 8 on a separate sheet of paper. You can check your answers at any time by hovering over the pink boxes.

Logarithms

Evaluating Logarithmic Expressions

• Khan Academy: Evaluating Logarithms• http://www.khanacademy.org/math/algebra/logarithms-t

utorial/logarithm_basics/e/logarithms_1• This is a problem set of practice in evaluating

logarithmic expressions. Try to get 5 questions in a row correct. In working these problems, remember that a log is determining a power of the base.

Logarithms

Evaluating Logarithmic Expressions

• Rewriting Expressions Quiz• http://www.mhhe.com/cgi-bin/netquiz_get.pl?qfooter=/u

sr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_7fq.htm&afooter=/usr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_7fa.htm&test=/usr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_7q.txt&answers=/usr/web/home/mhhe/math/precalc/barnettpc2/student/olc/quizzes/chap4_7a.txt&email=0

• This 18 multiple-choice question quiz will help you rewrite expressions between log and exponential form. Remember to hit “submit” at the bottom when done.

Logarithms

Graphs of Logarithmic Functions

Big Ideas: The graph of every logarithmic function has a vertical asymptote and a limited domain; however, the range is the set of real numbers. Using the function f(x) = logbx for some base b as the parent function, we can perform transformations on the function to get a new graph. Transformations include horizontal and vertical shifts (called translations), horizontal and vertical scale changes, and reflections about the x- or y-axis. Such transformations can change the domain and other properties of the logarithmic graph.

Logarithms

Graphs of Logarithmic Functions

• Khan Academy: Graphs of Logarithmic Functions• http://www.khanacademy.org/math/algebra/logarithms-t

utorial/logarithm_basics/v/graphing-logarithmic-functions

• This 9 minute and 10 second video has an example of how to graph log functions by hand. In particular, the parent function y = log5x is graphed. Pay attention to how x-values are picked to have “clean” y-values. Note the comment around 9:00 about how the graph never quite touches the y-axis. This is called a “vertical asymptote.”

Logarithms

Graphs of Logarithmic Functions

• Purple Math: Graphing Log Functions• http://www.purplemath.com/modules/graphlog.htm• This lesson shows how to graph the “parent” function of

a log as well as how to transform log functions to get different graphs. Page 1 shows how to graph the parent function y = log2x by hand. The bottom half of page 2 and all of page 3 show how adding/subtracting or multiplying/dividing by real numbers change the graph.

Logarithms

Graphs of Logarithmic Functions

• YouTube: Graphing Logarithms• https://www.youtube.com/watch?v=GT6AYjgoFco• This 13 minute and 57 second video walks through the

graph of y = log2x by commenting a lot about its relation to the exponential function y = 2x. At 8:20, he quickly discussed how you can use the graph of y = log2x to graph the transformed function y = log2(x - 3). Note that this shifts the graph right 3 units. Also at 10:30, he mentions some special properties of log graphs that you should note, especially the vertical asymptote.

Logarithms

Graphs of Logarithmic Functions

• Math Warehouse: Logarithm Applet• http://www.mathwarehouse.com/logarithm/graph/interac

tive-logarithm-graph-applet.php• Use this applet to view how different bases affect the

shape of a log graph. In particular, try changing the base in the drop down menu and observe when the graph is increasing and when it is decreasing. Try to select different bases, including those both less than 1 and greater than 1, including the special number e. Also think about how this graph relates to an exponential graph.

Logarithms

Graphs of Logarithmic Functions

• Analyze Math: Graphs of Logarithmic Functions• http://www.analyzemath.com/Graphing/GraphLogarithm

icFunction.html• Look at examples 1 and 2 to learn how to fully analyze

and graph transformed logarithmic functions by hand. Note how in the first example, the graph has moved two units to the left from the parent function. Also, in the second example, the graph has reflected over the x-axis due to multiplication by -3. There are matched problems with these two examples that you can try on your own.

Logarithms

Properties of Logarithms

Big Ideas: Logarithms have important algebraic properties that allow more complicated operations, such as multiplication, division, and exponentiation, to be simplified to addition, subtraction, and multiplication, respectively. Written symbolically:logb(xy) = logb(x) + logb(y)

logb(x/y) = logb(x) - logb(y)

logb(xy) = ylogb(x)

These properties allow us to either expand a logarithmic expression into multiple logs or condense multiple logs into one log expression.

Logarithms

Properties of Logarithms

• The Math Page: Laws of Logarithms• http://www.themathpage.com/aprecalc/logarithms.htm#l

aws• Read the first three statements about log properties, then

review examples 9 - 12, seeing how these three properties can be used. Then try problems 13 through 15, which you can check by hovering over the pink boxes. Proofs of the properties are included if you are interested. Also, be sure to read the comment about the Change of Base formula near the bottom of the page. It is written mathematically in the blue box at the bottom. You can try Problem 16 to test your understanding.

Logarithms

Properties of Logarithms

• Khan Academy: Introduction to Logarithm Properties

• http://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_properties/v/introduction-to-logarithm-properties

• This 9 minute and 15 second video starts with a review of writing an exponential in log form. At 1:37, the first property (involving multiplication and addition) is stated including a numerical example. At 6:00, the next property (involving division and subtraction) is stated with a numerical example to follow.

Logarithms

Properties of Logarithms

• Khan Academy: Operations with Logarithms• http://www.khanacademy.org/math/algebra/logarithms-t

utorial/logarithm_properties/e/logarithms_2• These practice problems will allow you to test your

understanding of how to use log properties to rewrite expressions. See if you can get 5 correct in a row!

Logarithms

Curve Fitting

Big Ideas: Given a set of data in the form of a scatter plot, one can perform a regression to determine a best-fit curve. Logarithms can be used to “normalize” the data. Specifically:-If taking the log of the outputs produces a linear function, then the original data is best modeled by an exponential function.-If a logarithm is applied to both the input and output thus producing a linear function, then the original data is best modeled by a power function.

Logarithms

Curve Fitting

• Curve Fitting (Free e-text from John H Heinbockel)• http://www.math.odu.edu/~jhh/chs5.pdf• This free electronic textbook contains a full chapter on

how logarithms can be used to fit a curve to statistical data. Note how log properties are used on page 167 of the text to express an exponential function as a linear function once logs are used. Also look at the two examples on pages 175 (exponential function) and 176 (power function).

Logarithms

Curve Fitting

• YouTube: Curve Fitting with Logarithmic Models• https://www.youtube.com/watch?v=KNfXzZ84Ie4• This 4 minute and 55 second video presents a worked

example of fitting a log function to real-life Richter Scale data from earthquakes. Unfortunately there is no audio available in the video.

Logarithms

Solving Logarithmic Equations Algebraically

Big Ideas: A logarithmic equation is one in which one or more logarithms are related by an equals sign. The goal of solving a logarithmic equation is to find the value (or values) that can be substituted for the independent variable that make the statement true. Properties of logarithms can be used to help solve an equation. When solving an equation it is important to be cautious with the solution set as logarithms are only defined for strictly positive inputs. Thus, some solutions may be extraneous since they do not fit the domain.

Logarithms

Solving Logarithmic Equations Algebraically

• Cool Math: Solving Log and Exponential Functions• http://www.coolmath.com/algebra/Algebra2/06Exponent

ialsLogs/10_equations.htm• These easy-to-follow examples show how logs can be

used to solve exponential equations. There are two opportunities for you to try similar problems and test your understanding. Note how logs of different bases are used at different times.

Logarithms

Solving Logarithmic Equations Algebraically

• Purple Math: Solving Log equations• http://www.purplemath.com/modules/solvelog.htm• This is a full lesson with many examples showcasing

how to solve different log equations algebraically.• Page 1 shows how to solve equations that have logs

on both sides. Note the last example on this page uses a familiar log property, turning addition to multiplication.

• Those on pages 2 and 3 are more complicated. Note the first two on page 2 use the definition of a log to change to exponential form.

Logarithms

Solving Logarithmic Equations Algebraically

• Khan Academy: Solving Log Equations• http://www.khanacademy.org/math/algebra/logarithms-t

utorial/logarithm_properties/v/solving-logarithmic-equations

• This 4 minute and 13 second video shows how all 3 of our important log properties can be used to solve an equation with logs on both sides.

Logarithms

Solving Logarithmic Equations Graphically

Big Ideas: Instead of solving logarithmic equations algebraically, such equations can be solved graphically. Either by hand or using a graphing utility, such as a calculator of software, the two sides of the equation can be separately graphed. The solution(s) to the equation are where the two graphs intersect.

Logarithms

Solving Logarithmic Equations Graphically

• YouTube: Solving a Log Equation• https://www.youtube.com/watch?v=nJb3uGGzKhQ• This video shows three different approaches to solve the

equation log2x + log2(x-2) = 3. Pay special attention at 2:50 where the solution x = -2 is not valid since it does not fit the domain of a logarithm (positive real numbers). Also note at 4:20 the Change of Base formula is used to be able to graph a logarithm with a base of 2.

Logarithms

Solving Logarithmic Equations Graphically

• TI Calc Help: Solving Logarithmic Equations Graphically

• http://ticalchelp.files.wordpress.com/2013/02/solving-logarithmic-equations-graphically.pdf

• This step-by-step guide shows how to use a graphing calculator (a TI-84 handheld) to solve the logarithmic equation log(x) + log(x-3) = 1. Note the base here is 10. You should also try solving this equation by hand and see that you get two solutions, one of which is extraneous.

Logarithms