UNIT 5Exponential and Logarithmic Functions!

OH YEAH!

Unit Essential Question• How are exponential and logarithmic functions related,

and how can they be represented graphically?

LESSON 5.1Exponential Functions

Lesson Essential Question (LEQ)• What is an exponential function and how can they be

represented graphically?

Solving Exponential Equations• Examples:

• 1)

• 2)

• 3)

• 4)

Sketching Graphs• Let’s sketch the graph of the following functions:

• Ex:

• Ex:

• Ex:

• Ex:

• Ex:

Even More Graphs!• Let’s sketch the graph of:

• Ex:

• Ex:

Real World Applications:• Compound Interest• Decay/Growth• Half-Life• Bloodstream• Appreciation/Depreciation• Inflation• Epidemics• Many more…

Homework:• Page 334-335• #’s 1, 3, 5, 7, 9, 13, 15, 17, 21, 30, 31a, 33a, 35

Bell Work:• 1) Solve for x.

Compound Interest:• Blue Table on Page 332

• A = Future Value• P = Principal• r = interest rate as a decimal• n = number of interest periods per year• t = number of years Principal is invested

Examples:• Ex: If you invested $2,000 dollars 10 years ago at 4.5%

that was compounded quarterly, what would be the value of that investment today?

• Ex: Mr. Kelsey needs to have $2,000,000 by the time he retires in 33 years. He plans to invest in a money market account that will return 5.25% per year. How much money will he need to invest right now to reach his goal?

Classwork/Homework:• Pages 335-336• #’s 37 – 42, 45-48

Bell Work:• The current average cost of gasoline per gallon in PA is

$2.79 and has been increasing at an average inflation rate of 3.75% per year. If this pattern holds true, what will be the cost of gas in 30 years?

LESSON 5.2The Natural Exponential Function

LESSON ESSENTIAL QUESTION• What is the natural exponential function and how can it be

used?

Important:

• How do we get this????

• The NATURAL EXPONENTIAL FUNCTION:

Continuously Compounded Interest

• A = Future Value• P = Principal• r = interest rate as a decimal• t = number of years Principal is invested

Law of Growth/Decay Formula

• If r > 0, then the quantity is growing.

• If r < 0, then the quantity is decaying.

Homework:• Pages 345-346• #’s 5 – 15 odds, 19-31 odds

Bell Work:• 1) If $12,000 is continuously compounded at 2.55% over

5 years and 3 months, what would be the amount of interest earned?

• 2) The population of PA in 1990 was 12 million. If the population is continuously growing at a rate of 1.05% per year, then what would the population be in 2020?

• 3) The amount A(t) of a certain radioactive isotope after t days is given as where is the initial amount of the isotope and the rate of decay is 0.225%. What was the initial amount if after 50 days, there was 893.6 grams left?

Small Quiz Monday:• You will need to know for tomorrow:

• Solve exponential equations• Compound Interest Formula• Word Problems• Solve natural exponential equations• Continuously Compounded Interest• Growth/Decay Formula • Word Problems!• WORD PROBLEMS!!!!!

Classwork/Homework:• Pages 345-347 #’s 6, 8, 12, 14, 20, 22, 24, 26a, 28, 32

• This assignment will be collected!

Bell Work:• 1) Solve:

• 2) Solve:

• 3) The half-life of a radioactive isotope is 400 years. If there are 600 mg of the isotope present at time t = 0, then the amount remaining after t years is given as

• A) How much of the isotope is remaining after 200 years?• B) Using the given equation, how long will it take for 75

mg of the isotope to be left?• C) Explain how you know the amount remaining after

2000 years without using the equation.

Examples:• 4) Mackenzie borrows $20,000 to purchase a new car.

Her loan is to be compounded monthly over 5 years at 4.5%. What will be her monthly payments?

• 5) The local population of koala bears has been decreasing since 1975 at a continuous rate of 6%. If the population is currently estimated to be 100 (in 2015), what would be the estimated population back in 1975?

• 6) Solve:

Bell Work:• 1) Colten begins a new job working for $15.25 an hour.

Every year he will get a annual 3% cost of living raise. If he works at this same job for 30 years, what will be his hourly wage?

LESSON 5.3Logarithmic Functions

Lesson Essential Question:• How are logarithmic functions related to exponential

functions and what are the different properties of logarithms?

Logarithm• if and only if

• x must be > 0• y must be a real number

• How does this compare to an exponential function?

Graphs of Logarithms• Let’s sketch the graph of a basic logarithmic function and

see how it compares to a basic exponential function.

• Sketch:

• and

Rewriting Logarithms• We can rewrite any logarithm in exponential form.

• .

• Let’s rewrite the following logarithms as exponents:

Rewriting Exponents• We can also rewrite exponents as logarithms.

• Ex:

• Ex:

• Ex:

Simplifying Logarithms• Simplify (if possible):

• Ex:

• Ex:

• Ex:

• Ex:

Bell Work:• Simplify:• 1) = ?

• 2) = ?

• 3) = ?

• 4) = ?

• 5) = ?

Properties of Logarithms

• Page 350 Blue Table

• Remember These Properties!!!!!

Solving Logarithmic Equations• Ex:

• Ex:

• Ex:

• Ex:

Common Logarithm• The most basic form of a logarithm:

• If the log does not have a specified base, it is assumed to be a base 10 log.

• Your calculator will only do base 10 logarithms!

NATURAL LOGARITHMS!!!!!!!!!• Just like there was a natural exponential function “e”, we

also have a natural logarithmic function! YES!

• Just like exponential functions and logarithms are inverses, the natural exponent and natural log are inverses!!!

• Ex:

Blue Table on Page 356• These are four properties of logarithms you should know,

as well as:

• If , then

• Ex: Convert to a base e expression.

Homework:• Pages 359 – 360 • #’s 2, 4, 10, 12, 14, 16 – 32 evens

Bell Work:• Be ready to ask questions on the homework!

• If not, begin working on the following assignment which is due Monday! (It will be collected!)

• Pages 359 – 360 #’s 1, 3, 13 – 31 odds

Bell Work:• Solve:• 1)

• 2)

• 3)

• 4) Solve for t.

• 5) Solve for k.

Graphing Logarithms• Lets create a table and sketch the graphs of the following:

• (graphing calculator)

• What do you notice about these graphs compared to exponential functions?

Shifting/Reflecting• The graphs of logarithms behave just like any other

functions.

• Lets sketch the graphs of some functions that will shift and/or reflect.

Class Examples:• Pages 362 – 363 #’s 52, 54, 60, 66, 70

Homework:• Page 360 #’s 51 - 69 odds

Bell Work:• The current population of Bloomsburg in 2015 is

approximately 12,000. It is projected to grow continuously in the future at a rate of 1.85%. How long will it take for the population of Bloomsburg to double in size?

LESSON 5.4Properties of Logarithms

Lesson Essential Question• What are the different properties of logarithms and how

are they used when simplifying exponential and/or logarithmic expressions?

Three More Properties of Logarithms• ORANGE TABLE ON PAGE 364

• These properties hold true for the common logarithm and the natural logarithm. (Blue Table Page 365)

Examples:• Using the laws of logarithms, rewrite the expression using

logs of x, y, or z.

• Ex:

• Ex:

• Ex:

Examples:• Using the properties of logarithms, rewrite each

expression as one logarithm.

• Ex:

• Ex:

Solving Logarithmic Equations:• Solve each logarithmic equation. Double check to make

sure the solution you found is in fact a solution!

• Ex:

• Ex:

• Ex:

• Ex:

Homework:• Page 370 #’s 1-33 odds

Bell Work:• Get out your homework from last night:• Page 370 1-33 odds

• Be ready to ask questions!

Quiz Tomorrow:• LOGARITHMS!• (remember that to answer some logarithmic problems you

need to know how to change to exponential form)• On the quiz:• Solving Logarithms• Word Problems• Properties of Logarithms• WORD PROBLEMS!!!

Class Work:• Pages 370 – 371 #’s 6, 8, 10, 12, 14, 18, 20, 22, 26, 34,

51, 52, 53, 54, 56

Bell Work:• 1) Armaan invests $25,000 into a mutual fund that has a

continuously compounded interest rate of 4.75%. How long will it take for Armaan to triple his investment?

• 2) Solve:

• 3) Solve:

LESSON 5.5Exponential and Logarithmic Equations

Lesson Essential Question• How can we change the bases of logarithmic and

exponential functions, and how do we use the special base formulas?

Example:• Solve:

• We can rewrite this in logarithmic form, but we still can’t solve it. Or can we?????????

Change of Base Formula:

• We can rewrite a logarithm with a “difficult” base as a quotient of two common logarithms or two natural logs.

• Ex:

• We can prove it!

Examples:• Solve each equation for x first, then approximate each to

two decimal places:

• Ex:

• Ex:

• Ex:

Solving an Exponential Equation • This problem is off the hook yo!!!!!!!!!!!!

• Ex: Solve for x, then approximate to the nearest 2 decimal places:

Homework:• Pages 381-382 #’s 1-31 odds only

Bell Work:• Solve for x.

Solving a Logarithmic Equation:• This pizzle is fo rizzle ( LOL):

• Ex: Solve:

Finding an inverse function:• Same type of problem that we solved in the bell work.

• Example: Find the inverse function of

• How could we use our graphing calculator to prove that the functions are indeed inverses?

Example using the Logistic Curve• A logistic curve is the graph of an equation in the form:• , where b, c, and k are constants, x represents• the time, and y is the population.

• Example: Assume c = 1.1244, k = 105, and x will be the time in days.

• A) Find the value of b if the initial population was 3.• B) How long will it take the population to reach 90?• C) Show that after a long period of time, the population of

this curve will become the constant k.

Class Work/Homework:• Pages 381-383 #’s 33 – 39 odds 49, 53b, 54, 55, 56a,

56c, 57

• If you need extra practice, try the evens (2 – 40)

Unit Test • Thursday and Friday we will be having our Unit 5 Test on

exponential and logarithmic functions.

• Here is a group of review exercises to try:• Pages 385 – 387• #’s 17 – 40, 45 and 46 (just find the inverse),• 47 – 55, 58, 61, 62, 66, 67