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6/19/2015 CCGPS Advanced Algebra

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CCGPS Advanced AlgebraExponential and Logarithmic Functions

Exponential and Logarithmic FunctionsDo you have a savings account or a loan for a car? Areyou concerned with nuclear power and radioactivity? Isstarting a business in your future, dealing with supply anddemand? Would a career in forensics, as a soundtechnician, studying earthquakes, or dealing withstatistics like population growth be of interest to you? Ormaybe you just want to know how your SAT scores stackup. These are some of the many instances whereexponential and logarithmic functions have value. Suchfunctions involve the use of a "power" of a number orexpression and are usually a measure over a period oftime.

Essential QuestionsHow can exponential expressions be rewritten fordifferent applications?What different properties of the function are shownin its graph?What is the relationship between exponential andlogarithmic functions?How are logarithms used to solve exponentialequations, and vice versa?

Module MinuteExponents (or powers) can be written asyou learned them in Algebra 1, as thenumber of times a base is used as afactor. Or they can be rewritten as alogarithm when you need to solve for thepower. The graphs of both show thedomain, the intercepts and the end

behavior of the function. Exponential and logarithmic functions are inverse function, where the x and the y are switched. Inmodeling realworld situations, we often need to solve these functions. The steps are similar to solving other algebraequations, but usually require changing from one form to the other. They are often used to calculate things like the time todouble, triple (an investment or a population), or halve (radioactive halflife) a quantity.

Key WordsExponential expressions expressions involving a variable as the base or the powerBase the expression being raised to a powerZeros the roots of a function, also called solutions or xintercepts.Intercepts points where a graph crosses an axisDomain the values for the x variableRange the values for the y variableAsymptotes vertical and horizontal lines where the function is undefinedExtrema maximums and minimums of a graphEnd behavior the rise or fall of the ends of the graphLogarithmic expression the inverse of an exponential expression

A handout of these key words and definitions is also available in the sidebar.

What To Expect

Exponential Rules and Graphs Handout AssignmentPopulation Growth Discussion



Properties and Graphing QuizSolve Equations Handout AssignmentSolve Equations and Models QuizGrowth and Decay ProjectExponential and Logarithmic Functions Test

To view the standards from this unit, please download the handout from the sidebar.

Properties of ExponentsBefore exponential functions can be explored, we must review the properties of exponents. Here is a list of the properties wewill be using.

Watch the video to see how these are used.

Another property is the rational exponents property.

The fraction in an exponent is the power divided by the root. A power of ½ is a first power and a square root; is asecond power and a cube root. You can do either the power first or the root first, though often it is easier to do the root first.

With a power like , you actually have three instructions. The negative means invert the base, the 3 means do the cuberoot, and the 4 means take the fourth power. You can do these three things in any order you choose.

Rollover these examples to see the radical expressions.



The same properties apply whether the exponent is afraction or an integer. See some examples here.

For more practice, explore Practice 1 and Quizlet inthe sidebar. If you need more explanation on therules, see Rational Exponents in the sidebar.

Graph Exponential FunctionsExponential functions are functions involvingexponential expressions with the variable in the power. There are two main types;

1. exponential growth functions in the form

2. exponential decay functions in the form

We will evaluate, find the yintercept, and graph exponential growth and decay functions.

To evaluate, substitute a number in for x and find y. To find the yintercept of the exponential function, set x=0. Finally,graph the exponential functions by making a table and plotting the points then drawing the curve. You will also see that keyfeatures like asymptotes and shifts are also used to analyze these functions.

Click here for an App to explore changes in exponential graphs. Click on "Hide keypad" on the left to see the sliders andinstructions.

For a more detailed view, click on App in the sidebar, turn on all of the features and move the sliders.

Here is a summary of the Key Features of growth and decay functions:

Example: Example:

Base > 1 or a>1 when

Base between 0 and 1 or 0<a<1

when

Domain all real numbers, Range isall positive real numbers.

Domain is all real numbers, Range isall positive real numbers.

No x intercept, y intercept is (0,1) No x intercept, y intercept is (0,1)

Smooth, continuous, no verticalasymptote

Smooth, continuous, no verticalasymptote

Horizontal asymptote is xaxis Horizontal asymptote is xaxis

Contains (1,1/a)(0,1)(1,a) Contains (1,1/a)(0,1)(1,a)

Increasing Decreasing

Larger base, steeper the graph Smaller base, steeper the graph

For more on these, see Exp Growth and Exp Decay in the sidebar.



Exponential Rules and Graphs Assignment

Select the " Exponential Rules and Graphs " Handout from the sidebar. Record your answers in a separatedocument. Submit your completed assignment.

Population Growth Discussion

It is now time to complete the " Population Growth " discussion. A rubric for your discussion is

located in the sidebar.

Population growth is an exponential growth function. Go to the following website to find out more about population growth:Population Growth Website.

1. What are some factors that influence population growth? 2. Can the population continue to grow without limits? 3. Why are clear charts of data important? 4. What are your feelings about population growth?5. Then research to find information on your state's population and its growth. Post the information that you find, including

at least one fact that your classmates haven't posted.

Introduction to LogarithmsIn this lesson, you will evaluate the logarithm and its properties. A logarithm is the inverse of the exponential function. (Youwill learn more about inverses in a later module.) This means that you can change a logarithmic function into an exponentialfunction and an exponential function into a logarithmic function. The way to do this is shown in the definition of a logarithm, or"log" abbreviated.

For all positive numbers b, with b≠1, and all positive numbers x



It is essential to know this relationship and how to convert from logs to exponents and exponents to logs.

Watch this video on changing form.

Here are some examples of simplifying logs:

See more examples and try more of these at Log Intro and Log Practice 1 in the sidebar.

Since "b" is the base and "y" is the exponent, the logarithm is an exponent. So the properties of logarithms follow from theproperties of exponents.

The base "b" can be number or variable. A common base is 10, since our number system is base 10. If fact, as a shortcut,the expression "log x" without a base shown, is understood to be base 10 and is called the common log.

Note: Another common logarithmic base is "e". The "natural log", is written as a short cut. The properties andrules are the same for this base. See Natural Log in the sidebar if you are interested in more on this common log.

Because logarithms are exponents, they match the rules for operations.

Product Rule

Quotient Rule

Power Rule

See how to use these rules to expand and condense logs.

For more, see Log Rules (first part of video), Log practice 2, and Log Practice 3 in the sidebar.



The graphs of logarithms are similar to those of exponents. But, since they are inverses, their graphs are reflections over theline y=x.

This means that domains and ranges are switched, and x's and y's are switched.

Click here for an App to explore changes in logarithmic graphs. Click on "Hide keypad" on the left to see the sliders andinstructions. Compare the key features of log graphs with those you found in the exponential graphs app on the previouspage. For more, click on Log App in the sidebar.

Here is a summary of the Key Features of log graphs:

Example: Domain: all positive real numbers.

Range: all real numbers.No y intercept, x intercept is (1,0)

Smooth, continuous, no horizontal asymptote

Y axis is the vertical asymptote

Contains (1,0), (a,1), (1/a, 1)

Decreasing when 0<a<1

Increasing when a> 1Smaller base, steeper graph

Let's look at graphs of logarithm.

Go to Graph Logs in the sidebar for more information.

Properties and Graphing Quiz

It is now time to complete the "Properties and Graphing" quiz. You will have a limited amount of time to complete yourquiz; please plan accordingly.

Solve EquationsIn this section, you will solve various types of exponential and logarithmic equations. Make sure you know all of yourexponential rules and your log properties, especially how to convert between exponents and logs.

An exponential property that allows us to solve some equations is "If , then . In order to use this exponential



rule, you have to make sure that each side of the equation has the same base. Then simplify any exponents and make themequal to solve the equation.

Here are some more complicated examples.

For more practice on these, go to Exp Practice in the sidebar.

If the bases are not the same you can change to logarithmic form or take the log of both sides. Then use the logarithmproperties to simplify and solve the equation.

For log equations, the first thing to try is to isolate the log, then change to exponential form.

Another method is using the log property "If . Remember, the bases must be the same and youmust have a single logarithm on each side of the equation. You may have to use your product, quotient or power rules here. Check your solution, because the domain of a log function is all positive real numbers. The expression in a logarithm cannotyield a negative, so there might be extraneous solutions.

Watch this Video Showcase to see examples of solving equations using these 3 methods.

You can also solve both types of equations with a calculator. Graph the expression on each side of the equal sign and findthe point(s) of intersection. But make sure you know how to solve these equations by hand or algebraically as you may beasked to show your work on assessments.

For more practice, see Log Rules and Equations in the sidebar.

Solve Equations Assignment

Select the " Solve Equations " Handout from the sidebar. Record your answers in a separate document. Submityour completed assignment.

ModelingNow that we have looked at different aspects of exponential functions, let's dive deeper into exponential growth and decay withreal world situations. Main uses for growth functions include interest on savings and loans, and population increases.

Exponential GrowthWhen examining exponential growth , a general case equation is:

y = total

C = original amount

r = rate (%) as a decimal



t = time

Exponential DecayMain uses for decay functions include depreciation and radioactivity. The general case for exponential decay contains all thesame information but is subtracted instead of being added.

For more examples, see Video Example and More Examples in the sidebar.

Now let's look at a real world use of logarithmic functions. Some other uses for logarithmic functions includesound intensity, ph levels and forensics.

Go to the sidebar to see more Log Examples.

Solve Equations and Models Quiz

It is now time to complete the " Solve Equations and Models " quiz. You will have a limited amount of time tocomplete your quiz; please plan accordingly.

Module Wrap UpModule ChecklistIn this module you were responsible for completing the following assignments:

Exponential Rules and Graphs HandoutProperties and Graphing QuizPopulation Growth DiscussionSolve Equations HandoutSolve Equations and Models QuizGrowth and Decay ProjectExponential and Logarithmic Functions Test

ReviewNow that you have completed the initial assessments for this module, review the lesson material with the practice activities andextra resources. Rewatch videos and visit the extra resources in the sidebars as needed. Then, continue to the next pagefor your final assessment instructions.



Standardized Test PreparationThe following problems will allow you to apply what you have learned in this module to how you may see questionsasked on a standardized test. Please follow the directions closely. Remember that you may have to use priorknowledge from previous units in order to answer the question correctly. If you have any questions or concerns,

please contact your instructor.

Final AssessmentsExponential and Logarithmic Functions Test

It is now time to complete the "Exponential and Logarithmic Functions " Test . Once you have completed all selfchecks, assignments, and the review items and feel confident in your understanding of this material, you may begin.You will have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your

test. Please plan accordingly.

Growth and Decay ProjectSelect "Growth and Decay Project " Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment. A rubric is available in the sidebar.

Documents

Exponential and Logarithmic Functions - Classroom Blogplanemath.weebly.com/uploads/1/3/5/1/13515714/ccgps_advanced... · logarithm when you need to solve for ... Exponential and logarithmic