Exponential Functions And Logarithms

Jody A. Carlisle

Day 1

- Students should be able to graph exponential functions and evaluate exponential expressions.
- Students should be able to use exponential functions to model real life situations such as the depreciation of a cars value.

On the board when the students enter the room are these two graphs:

Questions to be asked:
Which one of these graphs represents exponential decay? Why? Any other suggestions? What is the domain? Range? How did you get that?

These graphs are representations of exponential functions. The equation looks like f(x)=A^x where A is any real number greater than 1.

Students will have graphing calculators

Questions to be asked:
Why do you think A can not be 1? 0?

Why do you think A can not be a negative number?

The above graph shows -A^x but the graph of (-A)^x is only points on the graph.

Everyone put these equations into the y= screen: y1= 2^x y2=-2^x y3=(-2)^x
( you will need to use the carrot button).Now go to the table of values- What is different about y1 and y2 ? How is y3 different from y1 or y2?
If you have y=-2^x and let x=2 then what is y? y= (-1)2*2= -4
What about y=(-2)^x and let x=2 then what is y? y=(-2)(-2)=4
***In the homework tonight when it says to sketch the graphs of f(x)= -6^x then you will sketch the graph of y=-1*(6^x).

Now students will explore different types of exponential functions. They will be allowed to use any type of technology that is available. I will suggest that the students clear everything in your y= screen and graph y= 3^x ( you will need to use the carrot button). The questions that they will need to answer and hand in for each equation below is What is the Domain? Range?.
Now what would happen to this graph if:
y= 3^(x)-3 y= 3^(x+2)-5
y= 3^x+2 y= 3^(x-3)+5
y= 3^x-3 y= -3^x remember that this is -1*(3^x)
y= 3^x+2

Anyone have any idea why we use these types of functions?
One reason why we learn how to use these exponential functions is one day you may want to sell your car and you will need to know how to calculate the present day value ( so no one will rip you off) ?

We will us a formula that is almost the same as the one you used to find compound interest on yesterdays test.
A=P(1+(r/n))^nt this formula represents growth
we will use A=P(1-(r/n))nt but sense n will always be 1 b/c the value only decreases one time a yr.

A=P(1-r)^t where A is the value of the car
P is the price that you paid for the car
r is the rate in which it decreases
t is the age of the car

Transparency of the following:
Suppose that you purchased a new car for 8000 in 1991. If the value of the car decreases by 10% each year to 90% of its previous value what is the car worth today?

On the board
A= P(1-r)^t What is P?
A= 8000(1-r)^t What is r? r needs to be in decimal form A= 8000(1-.10)^t What is t? (1991-1997)
A= 8000(1-.10)^6
A= 4251

?? How can we make a graph of this function? Remember what an exponential function looks like
What about y= 8000(.90)^x graph this on your graphing calculator
What do you see? What should we set the window to be?
xmin=0 xmax=20 ymin= 0 ymax= 8000 yscl=1000
This graph will show us the value of the car at any age.

?? Use your trace key and find the value of the car if it is 15 yr. old? =1647
What is the value to 3 decimal places? =1647.129 (go to table)
Students will be able to explore with different values and years. They should get a good understanding how the independent and dependent varaible works.

Go over transparency #2
In 1990, you bought a television for $600. Each year, for t years, the value, v, of the television decreases by 8%. Write an exponential model that describes this situation.

V= 600(1-.08)^t

Next I am going to do a quick review. (Students to the board)
Positive Integer Exponent What does 4^3 mean? = 4*4*4 or an= a*a*a*...*a (n times)
Zero Exponent What is 5^0 ? = 1 or a0= 1
Rational Exponent 3^(1/4)= or a^(1/n)= 3^(5/4)= or a^(m/n)=
Negative Exponent 3^(-1/4)= or a(-1/n) =

Homework: p. 402: 1-4,7-11 odd, 13-20, 27-39 odd, 45-52, 53-60, 65,66

Assessment: Having students go to the board and answer the review questions above will give me a chance to see how the students learn or recall information. I allowed time at the end of class to start homework, and this gave me the opportunity to walk around the room to observe their style of learning.

Day 2

Objectives: Evaluate logarithmic expressions algebraically and with calculators
Convert between exponential and logarithmic form

I. Warm-up:
If a^n=a^x, then x=n. Ex. If 3^2=3^x, then x = 2.
Solve for x:
Evaluate: 3^-3, 5^-3, 10^0, 597^0, 10^-2, 10^3

II. Questions from last night's homework and completion of unfinished work from yesterday

III. Introduction of logarithms and converting between logarithmic and exponential form
A. Conduct whole-class discussion about evaluating 2^x=6...May need to note 2^2=4 and 2^3 = 9, so 2<x<3.
B. Briefly speak of John Napier, the inventor of logs.
C. When are we EVER going to use this?
Briefly explain about real-life applications of logarithms: decibels, Richter Scale, Carbon Dating.
D. Logarithms are just exponents!
E. Go through example 1 in textbook while writing some of the following on the board:
Logarithmic form Exponential form
log(2 )16=4 24=16
log 10=1 10^1=10
log(3 )1=0 3^0=1
log 0.1=-1 10^-1=0.1 or 1/10
log(4 )64=x _______
log(5) 125=y _______
log(7 )49=z _______

IV. Evaluating logarithmic expressions
log(4 )16
log(5 )1
log(9 )3
log(3 )-1
In whole class discussion, go through the answers to these problems. Ask students how they got their answers and call some to the board to write their methods down if necessary. Use exponential form, if necessary. Go back to examples already written on the board (log4 64=x, log5 125=y, and log7 49=z) and have students evaluate them for x, y, and z, respectively.

V. Special cases and quick things to point out
A. On the board or overhead, write:
log(a) 1=0 because a0=1
log(a) a=1 because a1=a
log(a) a^x=x because _____
Conduct a brief discussion as to what is to be written in the blank. (ax=ax)
B. Conduct a discussion as to why we cannot have base 1 nor take the log of anything less than or equal to zero. Include in the discussion the following examples:
Logarithmic form Exponential form
log(1) 500=x 1^x=500
log(3 )0 =x 3^x=0
log7 (-5)=x 7x=-5
C. Explain that log x can be written as log x. This is the common logarithm.
If time, show students on their graphing calculators that the "log" button means log10. Have them evaluate on the calculator: log 1000000 and log 106. Does 106 = 1000000? Explain that base 10 is the base used most often in our lives.

VI. Change of base formula and wrap up.
A. Go back to the problem 2^x=6 and explain/conduct discussion on how to solve. Confirm why we take log2 of both sides. log(2) 2^x=log2 6. So, x= log2 6.

B. Observe with students that there is no way to enter base on many calculators (specifically, the TI-82). Write the formula on the board for students to use to find x:

So, log(2) 6 = log 6/log2 ª 2.585. Go through the formula with students.

Homework Assignment:
In textbook, p. 409: 1-5, 11-14, 15-33 odd, 37-39, 41-55 odd. Due next class day (Monday, Feb. 17). See attached.
On homework, students MUST copy the problem down and show some work. Homework will be graded A, B, C, F, or 0 based on effort.

Day 3

Graphs and applications of logarithms

Goals: Reinforce evaluation of logarithmic expressions algebraically and with calculators
Reinforce conversion between exponential and logarithmic form
Graph simple logarithmic functions
Use logarithms in a real-life situation

I. Warm-up

log(6) 36=x
log(3) 0=x
log(2) (-4)=z
log(2) 8=x
log(5) 125=x
log(3) x=4
log(2) (x+4)=5

II. Questions from last night's homework and completion of unfinished work from yesterday

III. Graphs of basic logarithmic functions
With your graphing calculators graph y=log(2) x

B. Conduct whole class discussion of how to graph this and other logarithmic functions such as log x, log(5 )x. Use T-tables, graphing calculator and dry-erase graph board. Have student volunteer show how to come up with the graph without graphing it on the calculator. Be sure to note why we cannot start the T-table with x=0 (When will 2^y=0?). Students should notice what happens to the graph as base increases. Change of base formula may need to be noted: . To make the T-table, exponential form will likely be used.
C. On graphing calculator, graph alog (x+b) +c, varying a, b, and c. Conduct a class discussion about how varying these values affects the graphs.

On the overhead you will see that the top curve is when you add c, and the bottom curve is when you subtract c.

IV. Noticing inverse relations
A. With students, graph y=log x on graphing calculators. (Make sure students know this is logarithm base 10)
B. y=10^x. Notice anything?
C. y=x. Explain that these two functions are inverses of each other.

Real World Applications

Slope of a beach, s, is related to the diameter, d, of the sand particles on it by this equation:
s=0.159+0.118log d.

Have students work in groups of 2 to find the slope when d=0.25. Have student volunteer work it out on board. (Answer is roughly 9/100). Teacher should sketch a rough graph of what the slope of the beach looks like. The students can also use the trace key to find a solution. Tell the students to get as close to .25 as possible.

B. Have students evaluate d=0.125 (fine sand) in same groups. Have student volunteer graph what the slope of the beach looks like.

Homework Assignment:
1. In textbook, p. 409-411:6, 61-66, 69-77 odd, 79-83, 88-90. Due tomorrow. (Tuesday, Feb. 18).
2. For extra credit, Mixed Review in textbook p. 412. Due in four class days. (Friday, Feb. 21)
On homework, students MUST copy the problem down and show some work. Homework will be graded A, B, C, F, or 0 based on effort. Evaluation scheme for the Mixed Review has yet to be determined.

Properties of Logarithms

Day 4

Goals/ Objectives:
-Students should be able to use properties of logarithms.
-Students should be able to expand and condense logarithms.

Lesson Plans:

When I introduced log properties I wanted the students to make a connection with something that they already know; therefore, I had the following warm up problems on the board:

1- 4^3 * 4^2 = 4*4*4*4*4= 45 2- 2^4 * 2^5 = 2*2*2*2*2*2*2*2*2= 512
3- 3^6/3^4 = (3*3*3*3*3*3)/(3*3*3*3)= (6-4) 32= 9
4- 7^2/7^5 = (2-5) = 1/73
5- (7^3)^2= (7*7*7)^2 = (7*7*7)(7*7*7)= 76
My goal for the students to come up with the following on their own.
Therefore a^na^m = a^(n+m) (a^n)^m = a^nm a^n/a^m = a^(n-m)

Log properties are formed the same way. This is still on the board:
a^na^m = a^(n+m) (a^n)^m = a^nm a^n/a^m = a^(n-m)

?? What do you think loga^(nm)? ***loga means log base a

Let's prove that loga (nm) = loga n + loga m
Proof: Let loga m= x and loga n= y.
a^x= m and a^y= n Change to exponential form.
nm= a^xa^y Find the product of m & n.
nm= a^(x+y) Multiplying same base means adding exponets
loga nm= loga a^(x+y) Change to logarithmic form.
loga nm= (x+y) b/c loga a^x= x
loga nm= loga n + loga m Substitute for x & y.
Therefore loga nm = loga n + loga m
--Everyone get your calculators out and lets see if this is true.
The first 3 rows see what log(8*32) , and the last 3 rows see what log8+ log32?
Has anyone got an answer? 2.408 2.408 Are they the same?

?? What about loga m/n? ***loga means log base a

Let's prove that loga m/n = loga m- loga n together
Proof: Let ax= m and ay= n
loga m =x and loga n =y Change to exponential form
a^x/a^y = m/n Find the quotient of m/n.
a^(x-y)= m/n Property of exponents
loga a^(x-y) = loga (m/n) Property of equality for log functions.
x-y = loga (m/n) Definition of inverse function.
loga m - loga n = loga (m/n) Substitute for x and y.
Therefore loga m/n = loga m- loga n
-- Lets see if this property holds.
The first 3 rows see what log(27/3), and the last 3 rows see what log27- log3?
Has anyone got an answer? .954 .954 Are they the same?

?? Can any one guess what the loga n^m ?
loganm = m logan
(logan)^m (loga n) is an exponent to a power m, so you multiply them together.
(an)^m = n is the exponent to a power of m, so you multiply them together.
example: log10^2=2 or 2 log 10=2
Let's try this property out. The first 3 rows see what log8^4, and the other 3 row see what 4 log 8? 3.612 3.612 Are they the same?

Without your calculators lets do some more examples:
Let log2= .301 and log3= .477
?? What is log2/3? log 2/3= log2- log3
.301- .477 = -.176
What is log6? log6= log (2*3)
= log2+ log3
=.301 + .477 = .778
What is log9? log9= log3^2
= 3 log2 = 3(.301)= .954

Now, how can I rewrite log7x3?
= log7+ logx^3 Is there anything else that can be done?
= log7+ 3 logx This is called Expanding the expression.

What about logx- log3?
= logx/3 This is called Condensing the expression.

-- log 3xy^2?
= log3 +logx + 2logy
Is this condensing or expanding? How do you know?

-- log2- 2 logx?
= log2 - logx2= log(2/x2)

Being able to expand & condense helps to solve log functions
logx- log3=2 log7x^3= 2.345
log(x/3)= 2 log7 + 3 logx= 2.345
(x/3) =10^2 .8 + 3 logx= 2.345
x/3= 100 3 logx= 1.5
x= 300 logx= .5
x= 10^.5= 3.162

Homework: p. 416: 1-4,5,7,16-36 even
Assessment: I will allow time at the end of class to assess the students as they work on their homework.

Day 5

8.3 Properties of Logarithms

Goals/ Objectives:
-Students should be able to solve real- life problems using logarithms.
-Students will learn how to evaluate the intensity of earthquakes.
-Students will examine the patterns of decibels and intensity as they relate to sound.

Lesson Plans:
On the overhead when the students enter the room:
Expand the following problems: Condense the following problems:
log16x log6- log3/2
log6/5 10 logx+ (2/3)log64

**Students will answer the questions at the board.

?? Any questions?

Today we are going to us logarithms to solve problems

?? Has anyone ever felt an earthquake?
-- Transparency of the Richter Scale
This is a Richter scale. It is used to measure the strength of an earthquake. Each increase corresponds to a ten- times increase in intensity. For example, if an earthquake registers 8 on the Richter scale is ten times as intense as the one registering 7.

The formula that is used for the Richter scale is R= logI where R is the magnitude, I is the intensity ( intensity is a measure of the wave energy of an earthquake per unit of area).

How would we find the intensity of the Great San Francisco Earthquake in 1906, R= 8.3
R= logI 8.3 = logI How can we rewrite this into Exponential form?
I=10^8.3 = 199,526,231

Let's try another one. How would we find the intensity of the San Francisco Bay Area Earthquake in 1989, R= 7.1?
R= logI 7.1= logI How do we rewrite this one?
I= 10^7.1 = 12,589,254

If the intensity, I, is 92,523, what is R?
R= logI R= log 92,523 = 4.966 = 5
Another application that we are going to talk about today is Sounds and Decibels.

--Transparency of the Decibel Scale
This is a scale that represents different everyday sounds and the level of that sound.
Sounds travel by waves. These waves are picked up by our ears.
?? According to the chart what is the level of this classroom?

This worksheet that I am handing out is due by the end of the class. Make sure you put your name on the worksheet. Everyone needs to split up into groups of 3 and one group will have 4.




For the following questions us the formula:

B= 10 log (I/I(0))
where B is the sound level in Decibels,
I is the intensity of the sound in watts per square centimeter,
I(0) is an intensity of 10^-16 watts per square centimeter.

1- What is the sound intensity in a conversation?

2- What is the sound intensity with the decibel level of 90? Where are you when you experience this level of sound?

3- What is the sound intensity of a quiet room?

4- Choose 3 additional sounds you hear daily and estimate where you would place them in the scale shown on the transparency.

Homework: p.416-418: 40-46 even, 47-51 odd, 61,63, 68- 74.

Assessment: I will be walking around watching the students work and making sure that they are working. I will be looking for the leaders of the group so that we will be able to group them differently on the next lab. See rubric for evaluation for the homework.

Day 6

8.3 Properties of Logarithms (T-I 82 Lab)

Goals/ Objectives:
-Students should be able to use their calculator to do logarithms functions.

Lesson Plans:
On the board when students come into class:
(Students to board)
1-Is the graph of y= 3(.25)^x exponential growth or exponential decay.
2-Use a calculator to evaluate log3 to three decimal places.
3-Evaluate log(5)6 following to three decimal places using the change-of-base formula.
4-Expand log(x^3/y^4)
5-Condense log3+ 2logt

Today is a lab day.

Mini- Lab 38 Logarithmic Functions 2

Solve for x: 2=log(x-7)- log (x+3)

Sketch the graph of 2= log(x-7)- log (x+3)

What happens around -3?

What is the domain and the range of log(x-7)- log (x+3)?

Solve for x: 2= log(x-7) + log (x+3)

Sketch the graph of 2= log(x-7)+og (x+3)

What happens around -3?

What is the domain and the range of log(x-7)+ log (x+3)?

This lab is due at the end of the class, and no one should have problems with finishing.
Everyone needs to stay in their desk and do the lab on their own.

Homework: none

Assessment: I will be walking around helping and observing the students. The lab be graded on effort like the homework.

Day 7

The natural base e.

Goals: Students should know how to use the number e as a base of an exponential function.
Students will use the natural base e in real-life applications

I. Warm-up

Identify the following functions as representing exponential growth or exponential decay: f(x)=3(0.25)^x g(x)=2(2.1)^x

III. Introduction of the natural base e
A. Briefly mention Leonhard Euler, for whom e takes its name.
B. Show/describe that e=1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ...
ª 2.718281828459
(may need to review factorial notation)
C. Review that x and y are examples of variables, but e is a number. If time, show that it is irrational.
D. Have students evaluate eb for several values of b.
E. When are we EVER going to use this? Natural base e applies to interest rates on accounts, air pressure, population, and radioactive decay.

IV. Simplifying natural base expressions
Briefly go over the rules of exponents which apply to e just as they apply to any number or variable.
(e^2)(e^3) = e^2+3 = e^5
(6e^2)/(2e^1) = 3e^2-1 = 3e
e^-4 = 1/e^4
(2e^-1)^2 = 2*2^(e-2) = 4^(e-2) = 4/e^2

V. Graph of the base e function
A. Have students graph y=2^x and y=3^x, zooming in to see the graphs more clearly.
B. Next, have students graph y=e^x. Ask them about where the graph of y=e^x lies and why they think this must be so. 2<e<3.

C. Go over some shifts of the y=ex curve. Graph y=Ce^bx +d + a varying a, d and C at first, then b last. Lastly, make b negative to yield y=Cebx . Let students discover that if b is negative, the function represents decay, and if b is positive, the function represents growth.
D. Go through some examples of exponential decay and growth:

VI. Using base e to growth of money with a continuously compounding interest rate.
A. Recall with students the formula for compounding interest n times per year: A=P(1+r/n)^nt. Prompt students for what the variables mean: A - balance, P-initial principle, r- annual interest rate, n- number of times per year that interest is compounded.
B. What if interest were compounded continuously, that is, roughly every second?
A=Pe^(rt) where all the variables remain the same except we have no n.
C. Using the graphing calculator and previous formulas, go through second example with students:
You just deposited $200 in a savings account which has an APR of 5.0%. If interest were compounded continuously, what would the balance be after five years ($256.81)? What if interest was compounded daily? ($256.80) Our y= screen should look like this:

Now graph the function

Use your trace key to find x=5

Give other values for the students to find, so they will use the trace key.

D. If time, go through more examples with students. Initial principle = $5000, interest = 3%. Compare compunding semi-annually and continuously.
Note that with smaller initial deposits, the continuous compounding does not differ much from compounding daily.

Test Tomorrow

Homework Assignment:
In textbook, p. 423-425: 1-4, 9-21 odd, 25-32, 39-41, 45-52, 58-60.
On homework, students MUST copy the problem down and show some work. Homework will be graded A, B, C, F, or 0 based on effort.

Day 8

Review Day

Goals/Objectives: Students should be able to exhibit knowledge of the material covered in the past 8 days (and, of course, all the days before).

Conduct the following review (which Parallels the test) and then open the class to any other questions

Compare the following graphs?
y= 5x+7 y= 5(x+7)

Expand the following:
log(3/a) logx^4

Condense the following:
log3+ logx 7logb

The value of the new car you bought in 1990 for $17800 decreases by 16% each year. What will the car be worth in 1998?

?? Does anyone have any other questions from your homework?

Evaluate to 3 decimal places
log(2) 32 log99

Solve for x
log(x) 64= 6 log2x + log25= 4

Is f(x)= 10.2e^.04t an example of exponential growth or decay?

$600 is deposited in an account that pay 7% annual interest compounded continuously. Use the formula A= Pe^rt to find the balance after 10 years.

?? Does anyone have any other questions from your homework?

Homework: Study for test!!!

Assessment: I will be walking around helping the students that have any questions.

Day 9 Test Day

Test A

Name__________________________ Date_________

1. Evaluate e^2 to three decimal places.

2. Compare the graphs of f(x)=3^x and g(x)=3^x+1.

3. Craig and Jessica bought a new car in 1984 for $16,000. The car's value decreased by 13% each year. What was the car's value in 1992?

4. Solve for x. log 8 - (1/3)logx = log 2

5. Expand the expression log(4) 5x^2.

6. Condense the expression log(4) x - log(4 )5.

7. Solve for x. log(4) (1/16) = x

8. Which of the following is undefined?
log(3) 9, log(3) 1, log(3) (-3)


9. Evaluate log(6) 30 to three decimal places.

10. Solve for x. log7 2401 = x

11. Use the grid above to sketch the graph of f(x) = log(4) (x+1).

12. Evaluate 6^e to three decimal places.

13. Is f(x) = 13.7e^-0.04t an example of exponential growth or exponential decay?

14. $1000 is deposited in an account that pays 7% annual interest compounded CONTINUOUSLY. Use the formula A = Pert to find the balance after 10 years.

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