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2.1 EXPONENTIAL AND LOGISTIC FUNCTIONS

Learning Targets: 1. Recognize exponential growth and decay functions 2. Write an exponential function given the y-intercept and another point (from a table or a graph).

3. Be able to define the number e 4. Use transformations to graph exponential functions without a calculator. 5. Recognize a logistic growth function and when it is appropriate to use.

6. Use a logistic growth model to answer questions in context.

So far we have discussed polynomial functions (linear, quadratic, cubic, etc.) and rational functions. In all of these functions, the variable x was the base. In exponential functions, the variable x is the exponent.

Exponential equations will be written as _, where a = .

Exponential functions are classified as growth (if b > 1) or decay (if 0 < b < 1).

Example 1: Determine a formula for the exponential function whose graph is shown below. y

(2, 6)

(0, 2)

x

Example 2: Describe how to transform the graph of 𝑓(𝑥) = 3𝑥 into the graph of (𝑥) = 3𝑥+2 − 1 .

Example 3: Sketch the graph of the exponential function.

a) 𝑔(𝑥) = 3−𝑥 b) 𝑔(𝑥) = −3𝑥 + 2 =

The Number e

Many exponential functions in the real world (ones that grow/decay on a continuous basis) are modeled using the base of e.

Just like 𝜋 = 3.14 , we say e = 2.718 . We can also define e using the function (1 +1

𝑥)𝑥

as follows:

As 𝑥 → ∞, (1 +1

𝑥)𝑥

→ 𝑒

Try convincing yourself that this function approaches e using the TABLE function of your calculator.

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Do you think it is reasonable for a population to grow exponentially indefinitely?

Logistic Growth Functions … functions that model situations where exponential growth is limited.

An equation of the form or .

where c = .

The graph of a logistic function looks like an exponential function at first, but then “levels off” at y = c. Remember from

our Parent Functions in chapter 1, that the logistic function has two HA: y = 0 and y = c.

Example 4: The number of students infected with flu after t days at Springfield High School is modeled by the following function:

𝑃(𝑡) =1600

1+99𝑒−0.4𝑡

a) What was the initial number of infected students t = 0?

b) After 5 days, how many students will be infected?

c) What is the maximum number of students that will be infected?

d) According to this model, when will the number of students infected be 800?

Example 5: Find the y–intercept and the horizontal asymptotes. Sketch the graph.

a) 𝑓(𝑥) =9

1+2(0.6)𝑥 b) 𝑓(𝑥) =

8

1+4𝑒−𝑥

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2.2 EXPONENTIAL AND LOGISTIC MODELING

Learning Targets:

1. Write an exponential growth or decay model f ( x)

= ab x

and use it to answer questions in context.

Exponential Growth Factor… b = 1 + % rate (as a decimal)

Exponential Decay Factor… b = 1 – % rate (as a decimal) 2. Write a logistic growth function given the y-intercept, both horizontal asymptotes, and another point.

Example 1: The population of Glenbrook in the year 1910 was 4200. Assume the population increased at the rate of 2.25%

per year.

a) Write an exponential model for the population of Glenbrook. Define your variables.

b) Determine the population in 1930 and 1900.

c) Determine when the population is double the original amount.

Example 2: The half-life of a certain radioactive substance is 14 days. There are 10 grams present initially.

a) Express the amount of substance remaining as an exponential function of time. Define your variables.

b) When will there be less than 1 gram remaining?

Example 3: Find a logistic equation of the form 𝑦 =𝑐

1+𝑎𝑒−𝑏𝑥 that fits the graph below, if the y-intercept is 5 and the point

(24, 135) is on the curve.

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2.3 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

Learning Targets:

1. Rewrite logarithmic expressions as exponential expressions (and vice-versa). 2. Evaluate logarithmic expressions with and without a calculator.

3. Graph and translate a logarithmic function.

Rewriting Logarithmic Expressions

A logarithmic function is simply an inverse of an exponential function. The following definition relates the two functions:

logb y = x if and only if bx = y

If you think of the graph of the exponential function y = bx , x could be any real number, but y > 0. These same

limitations on the variables are true for the logarithm function as well.

Example 1: Evaluate each expression without a calculator.

a) log5 125 b) log

7 1 c) log 1 81 d) log

8 2

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Two Special Logarithms

A logarithm with base 10 is called a logarithm and is written .

A logarithm with base e is called a logarithm and is written .

Example 2: Evaluate each expression without a calculator.

a) log√104

b) ln1

𝑒

A Consequence of the Inverse Properties of Logarithms:

blogb M = M

… and …

logb (bM ) = M

Example 3: Evaluate each logarithmic expression without a calculator.

a) log6 63

b) log 105

c) eln 5

d) 8log8 7

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Example 4: Graph the following functions without a calculator.

a) y = log3 x b) y = ln x

y y

x x

Example 5: Graph the following transformations of the two functions above without a calculator.

a) y = log3 (x – 3) + 1

y

b) y = ln (–x)

y

x x

Example 6: Graph y = log3 (2 – x) without a calculator.

y

x

6

log216

log416

log5125

log1/636

log1000

log1616

log401

log2(32)

ln e

eln6

7log 7 2

log44

ln √𝑒4

log x = 2

log2( ½ )

log327

ln e7

log449

log1/39

log2√2

log31

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Graphing Logarithms Name:______________________________________

Directions: Identify the domain, range, horizontal and/or vertical asymptotes.

Then sketch the graph. State the transformations from the parent function.

1. y = log6 (x – 1) – 5 2. y = log5 (x – 1 ) + 3

3. y = – log5 (x + 1) + 1 4. y = ln( – x) +2

5. y = log2 (x + 2) – 2 6. y = – ln(x – 4) + 4

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Practice with Logarithms Worksheet

I. Express the equation in exponential from.

1. log525 = 2 2. log82 = 1/3

II. Express the equation in logarithmic form.

3. 53 = 125 4. 8-1 = 1/8

III. Evaluate the expression. (WITHOUT A CALCULATOR!)

5. (a) log636 (b) log981 (c) log7710

6. (a) log3(1/27) (b) log10√10 (c) log50.2

7. (a) 2log2 37 (b) 3log3 8 (c) 𝑒ln √5

8. (a) 𝑒ln 𝜋 (b) 10log 5 (c) 10log 87

IV. Use the definition of the logarithmic function to find x.

(WITHOUT A CALCULATOR!)

9. (a) log5 x = 4 (b) log 0.1 = x

10. (a) log4 2 = x (b) log4 x = 2

11. (a) log x1000 = 10 (b) log x 25 = 2

V. Find the domain of the function.

12. 𝑓(𝑥) = log10(𝑥 + 3) 13. 𝑓(𝑥) = log5(𝑥 − 8)

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VI. Graph the function, not by plotting points, but using transformations. State the

domain, range, and asymptote.

14. 𝑓(𝑥) = log2(𝑥 − 4) 15. 𝑦 = log3(𝑥 − 1) − 2

16. Draw the graph of y = 4x, then use it to draw the graph of y = log 4 x.

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2.4 PROPERTIES OF LOGARITHMS

Learning Targets: 1. Use properties of logarithms to expand a logarithmic expression. 2. Use properties of logarithms to write a logarithmic expression as a single logarithm.

3. Evaluate logarithms with bases other than e or 10 using a calculator.

Properties of Logarithms … these properties follow from the properties of exponents

3 Rules of Logarithms: 1. log𝑏(𝑀𝑁) = log𝑏(𝑀) + log𝑏(𝑁)

2. log𝑏 (𝑀

𝑁) = log𝑏(𝑀) − log𝑏(𝑁)

3. log𝑏(𝑀𝑘) = 𝑘 ∙ log𝑏(𝑀)

Example 1: Expand each logarithmic expression.

a) log (xy3 )

b) ln (𝑥2 √𝑞3

𝑦√𝑡)

Example 2: Condense each logarithmic expression into a single logarithm.

a) 3 ln 𝑥 −1

2ln 𝑦 + 5 ln 𝑧 b) 3 ln 𝑥 − (

1

2ln 𝑦 + 5 ln 𝑧) c)

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3log𝑏 27 − 2 log𝑏 9

Your calculator (unless you have the new TI-84+ operating system) will only evaluate a logarithm with base 10 or e.

If you need to evaluate a logarithm with a different base, you must use the change of base formula.

Change of Base Formula: log𝑏 𝑎 =log 𝑎

log 𝑏 or

ln 𝑎

ln 𝑏

Example 3: Evaluate each logarithmic expression using your calculator.

a) log7 5

b) log0.4 8

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Expand:

ln (√𝑥3

√𝑦3)

Simplify:

(1

3) log2 𝑥 − log2(𝑦𝑧3)

Simplify:

3 ln(𝑥3𝑦) + 2 ln(𝑦𝑧2)

Use change of base to evaluate: log0.2 29

Use change of base to evaluate:

log5(𝑐 − 𝑑)

Expand:

log (1000𝑥4

3𝑦)

Expand:

(1

2) log3 (

𝑥4

𝑧 + 3)

Simplify:

3 ln (𝑒1−𝑥

2)

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49. Given 𝑓(𝑥) = log(𝑥2) Graph the function on your calculator and analyze it for the following:

Domain:

Range:

Continuity:

Increasing/Decreasing Behavior:

Asymptotes:

End Behavior:

Problem A

Let log𝑏 2 = 𝑥, log𝑏 3 = 𝑦, and log𝑏 5 = 𝑧.

i) What is the value of log𝑏 50 in terms of x, y, and z.

ii) What is the value of log𝑏 3000 in terms of x, y, and z.

Problem B

Are log2 16 and log4 64 equal? Why or why not?

Problem C Correct the error

Simplify log2(6𝑥)5

log2(6𝑥)5 = 5 log2(6 ∙ 𝑥) = 5 ∙ log2 6 + log2 𝑥

= 5 log2 6 + log2 𝑥

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2.5 EQUATION SOLVING AND MODELING

Learning Targets: 1. Solve exponential and logarithmic equations.

When you solve an equation, you “undo” what has been done … addition to undo subtraction, multiplication to undo

division. Since exponents and logarithms are inverses of each other, it follows that in order to solve a logarithmic equation,

you can write it as an exponent to “undo” the logarithm, and if you are solving for an exponent, you write the equation as a

logarithm. NOTE: You can only switch between exponential and logarithmic forms when you have log b y = x or 𝑏𝑥 = 𝑦

Example 1: Solve the following equations:

a) 𝑒𝑥−2 = 19 b) 3 − 5(2)3𝑥 = −14

c) log2(𝑥) = 4 d) 7 − 3 log(𝑥) = −5

e) log 𝑥 + log(𝑥 + 21) = 2 f) log3(𝑥 + 4) − log3(𝑥 − 5) = 2

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2.6 MATHEMATICS OF FINANCE

Learning Targets:

1. Apply correct compounding interest formulas (either continuously or n times per year) 2. Use the FINANCE application on your TI-83 or 84 calculator to solve for unknown quantities.

Compounding Interest “n” Times Per Year

When the interest earned is added to the original amount and then interest is calculated on that interest (in other words, you

are earning interest on your interest), we say the interest is compounded. The formula for compounding interest n times

each year is given by

𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡

Where … A = new amount

P = original amount or Principle

r = the interest rate as a decimal n = the number of times a year interest is calculated t = the number of years

On your own: Consider the situation where n = 1.

a) Explain what this means in the context of compounding interest.

b) What does the compounding interest rate look like when n = 1?

Example 1: How much time is required for an investment to double in value if interest is earned at the rate of 5.75% compounded quarterly?

Compounding Interest Continuously

The formula for compounding interest continuously is . Notice in this formula, you need the

interest rate r and NOT the growth factor b.

Example 2: Find the value of an investment with a principle of $3,350 at 6.2% compounded continuously for 8 years.

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Using the FINANCE App On Your TI-83 or TI-84 Calculator

Your graphing calculator has a financial calculator application built into its programming called the TMV Solver. TMV

stands for Time–Value–of–Money. The application solves for one variable when given values for any four of the five

remaining variables. It is especially useful when you are making the same periodic payment (or investment) more than

once a year…i.e. car payment.

First we need to understand the 5 variables involved…Press APPS, Finance, TMV Solver to get started…

1. n = the number of compounding periods

2. I% = the annual percentage rate 3. PV = the Present Value of the account or investment 4. PMT = the payment amount

5. FV = the Future Value of the account or investment

6. P/Y = payments per year

7. C/Y = compounding periods per year…for our use C/Y = P/Y 8. PMT: END BEGIN means payment at the end or beginning of the

payment period.

NOTE: Enter any money received in as positive numbers and any money paid out as negative numbers.

Example 6: Sal opened a retirement account and pays $50 at the end of each quarter (4 times/year) into it. The account earns 6.12% annual interest. Sal plans to retire in 15 years. How much money will be in his account at that time?

Example 7: Kim wants to buy a used car for $9000. The bank offers her a 4-year loan with an APR of 7.95%. What will Kim’s monthly car payment be? Assume she makes her payment on the 1

st of each month.

Example 8: Compound Interest: $100 is deposited into a bank at 6% annual interest for 5 years. How much will the balance be at the end of the 5th year if interest is compounded a) annually? b) monthly? Example 9: When you were born your grandparents deposited $5000.00 into a special account for your 21st birthday. The interest was compounded monthly at 5%. How much will it be worth on your 21st birthday?

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Exploring Savings and Loans

Practice Problems

1. Jerry received a bonus at work of $2,400. He decided to put it in a savings account at 4.25% interest

compounded daily. What would be the value of the account in six years if no additional money was

deposited or withdrawn? SOLVE BY HAND and use FINANCE APP to check work, write all values

2. Betty will be attending medical school after graduation from college in five years. She estimates that

medical school will cost $50,000. A local bank is offering a 60-month CD at a rate of 5.5% interest

compounded monthly. Her parents have just received some inheritance money. How much of the

inheritance money must her parents invest in the CD to accumulate $50,000 at the end of the five years?

SOLVE BY HAND and use FINANCE APP to check work, write all values.

3. Ace Automotive is advertising 4.9% interest with no down payment on all “demo” cars. Richard selected a

red SUV that cost $23,250. If Richard financed the car for four years, what would be his monthly

payments.

SOLVE BY HAND and use FINANCE APP to check work, write all values.

4. Taylor wants to purchase a car that costs $32,500. If he wants to finance the entire amount for no more

than four years and have monthly payments no higher than $700, what is the highest interest rate that he

could accept? Use FINANCE APP, write all values.

5. Susan chose to finance the purchase of her house with a $120,000 mortgage at 7.70% APR. If the

mortgage is for 30 years, what are the monthly payments Susan must make on her mortgage? SOLVE BY

HAND and use FINANCE APP to check work, write all values.

6. How much time is required for a $1000 investment to double in value if interest is earned at the rate of

5.75% compounded quarterly? SOLVE BY HAND

7. James is investing $1500 for 20 years at a 6.25% annual interest rate. How much will be in the account if

interest is compounded daily? SOLVE BY HAND and use FINANCE APP to check work, write all

values.

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Which Car Can I Afford?

You want to buy a car. You go to a car dealer and start looking around. One of his first questions that you need to

consider is “How much money do you want your monthly payment to be?” You, as the consumer may be thinking

that a car loan with a lower monthly payment means that the car is cheaper. There are a variety of loan options

available so the length of the loan and the interest rates must also be considered.

Do you think that it is always a good idea to choose a loan with lowest monthly payment? Why is it necessary to

consider the length of time that you will be making payments on the loan? Explain below and justify your

reasoning:

To purchase a car for this activity, you must take out a car loan. You could buy a 2007 Mustang GT Premium

Convertible for 31,840 or a 2003 Mustang GT Premium Convertible for 19,120. Which loan offers the better value

and is affordable?

2007 Mustang GT Premium Convertible 2003 Mustang GT Premium Convertible

$31, 840 $19,120

Calculate the missing values for both a 3-year loan and a 5-year loan.

2007 Mustang 2003 Mustang

Table 1. 3-year loan at 6.5% 5-year loan at 7.15%

Number of Months

Interest (%)

Principal Value

Monthly Payment

Based on the data collected in Table 1, which car loan payment appears to be a better value? Explain your

reasoning.

You tell the car salesperson that you really were thinking of a lower monthly payment than either of the previous

figures. He says, “No problem, we can lower the 5-year loan payment by about $40.” You are quite relieved;

however your new loan will have a term of 6 years instead of 5 years. You now have an affordable payment but

how will this deal affect you financially in the future? What do you think?

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Being a student, you decide to purchase the cheaper 2003 Mustang. Complete the table below. In order to

calculate the total price (sum of all loan payments), multiply the monthly payment by the number of months. To

find the amount of interest paid, find the difference (subtract) between the principal value and the total price.

Table 2. 2003 Mustang 5 year loan at 7.15% 6 year loan at 8.2%

Number of Months

Interest (%)

Principle Value

Monthly Payment

Total Price

Interest Paid

Do you think that people need to understand how interest is calculated on the loans they borrow? Why is it

important?

Extension:

Now let’s try something else. Assume that you have decided to go with the 6 year loan at 8.2% because you need

the lowest payment. You just received an hourly raise at work and decide that you can afford to pay an additional

$30 each month on your car payment beginning with the first payment.

How do you think this will that affect the number of payments?

Instead of the loan taking 72 months it will be paid off in ___ months.

How much money would be saved in interest?

Making a Down payment:

Another option that can be done is to make a down payment, which is to pay a sum of money to the lender at the

beginning of the loan. Let’s say that you want to put down a $2000 down payment for the car. A car dealer’s loan

advisor may tell you …

“It won’t make a significant difference for your loan payments!”

Let’s investigate whether or not this statement is valid.

Using the 6 year loan data, apply a down payment of $2000. Notice that you will only need to borrow $17,120 for

the loan.

What would the new monthly payment be?

How much did this reduce the monthly payment?

How much money would be saved in interest?

Is the amount of money saved in interest “significant” to you? Who loses money when you make a down

payment?