Algebra 2 and
Trigonometry
Chapter 8: Logarithms
Name:______________________________
Teacher:____________________________
Pd: _______
1
Table of Contents
Day 1: Chapter 8-1/8-2: Converting an Exponential Equation into a Logarithmic Equation
SWBAT: Convert an Exponential Equation into a Logarithmic Equation
Pgs. 3 – 11 in Packet
HW: 326 – 327 #3 – 73 (every other odd)
Day 2: Chapter 8-3: Properties of Logs
SWBAT: Learn the properties of logs.
Pgs. 12– 18 in Packet
HW:
Page 331 – 332 #15 – 53 (every other odd)
Day 3: Chapter 8-3: Review of Properties of Logs
SWBAT: Learn the properties of logs.
“REVIEW DAY”
QUIZ on Day 4
Day 4: Chapter 8-6/8-7: Solving Equations with Logs
SWBAT: Solve Equations using Logs
Pgs. 19 – 23 in Packet
HW:
Page 343 #3 – 14
Page 346 #3 – 17 (odd)
Day 5: Chapter 8-5: The Natural Log
SWBAT: Solve Exponential Equations with like and unlike bases
Pgs. 24 – 30 in Packet
HW:
Pgs. 31 – 33 in Packet
Day 6: Review
SWBAT: Solve Log Equations
Pgs. 34 – 35 in Packet
HW:
Booklet (not in this Packet)
QUIZ on Day 7
Day 7: Exponential Growth and Decay
SWBAT: Solve applications of Exponential Equations
Pgs. 36 – 41 in Packet
HW: Pgs. 42 – 44 in Packet
Day 8: More Applications of Exponential Equations
SWBAT: Solve applications of Exponential Equations
Pgs. 45 – 53 in Packet
HW: _________________
Day 9: Inverses and Graphs of Logarithmic Functions
SWBAT: Graph Logarithmic Functions
Pgs. 54 – 59 in Packet
HW: Pgs. 60 – 62 in Packet
2
Name _________________ Date ______ Per. _____/Ms. Williams
Day 1 – Converting an Exponential Equation into a Logarithmic
Equation
SWBAT:
Convert an Exponential Equation into a Logarithmic Equation
Warm - Up
Solve the following system of equations algebraically:
9x 2  y 2  9
3x  y  3
3
Concept 1: Converting from Exponential to Logarithmic Form and Vice Versa
Until now, there was no way to isolate y in an equation of the form
.
You cannot take the “yth root” of something if that something isn’t a value.
The word “Logarithm” means “power.” When you see the function “log,” you should translate that into
“the power I raise…”
Example:
Log2
The power I raise
2 to to get
8
8
=
3
is
3
Example:
X
=
What
Log10
1000
The power
I raise 10 to
to get
is
10000
So,
=
4
OR
5
Concept 1:
How do we change between log form and exponential form?
Teacher Modeled
Student Try it!
6
Concept 2:
Simplifying a Log Expression
Rule:
Teacher Modeled
Student Try it!
7
Concept 3:
How Do we evaluate Common Logs?
Base with NO Exponent
Radical with NO index
√
X
means…….
Logarithms with a base
means…….
means…….
Logarithms with NO base
means…….
On the graphing Calculator
8
Evaluate each.
9
Challenge: Solve the following Equation
Summary/Closure:
Exit Ticket:
Homework: Page 326 – 327 #3 – 73 (every other odd)
10
11
Name _________________ Date ______ Per. _____/Ms. Williams
Day 2 – Properties of Logs
SWBAT: Learn the properties of Logs
Solve.
12
Algebra2/Trig: Logarithm Distribution Rules
Let M = 107 and N = 1012 . Then log M =______________ and log N = ________________
What is
? _________________________
(This means multiply M and N first, then determine the log of the product.)
The log of a product is the ____________ of the individual logs.
What is
( )?__________
(This means divide M by N first, then determine the log of the quotient.)
The log of a quotient is the ____________ of the individual logs.
Now, let M = 107 and c = 5. What is
____________?
(This means raise M to the power of c first, then determine the log of that answer.)
The log of a power : ____________ the power by the individual log.
Log Distribution Rules (Summary):
Product Rule
Quotient Rule
( )
Power Rule
Expanding Logarithms
Examples
1. log (a2b3c)=
log a2 + log b3 + log c=
2 log a + 3 log b + log c
3.
(
) =
2
log a – (log b3 + log c5)=
2 log a – (3 log b + 5log c)
2 log a – 3 log b - 5log c
(product rule)
(power rule)
2.
4.
(product/quotient rule)
(power rule)
(distribute)
(
)=
2
log a + log b3 - log c5 = (product/quotient rule)
2 log a + 3 log b - 5log c (power rule)
(
√
√
) =
(product/quotient rule)
(evaluate log 100)
** RECALL ROOTS ARE EXPONENTS
13
Concept 1: Expanding Logs
Expand each using the properties of Logs
1.
2.
3.
4.
5.
7.
Fdf
6.
8.
14
Concept 2: Condensing Logs
9.
10.
11.
12.
13.
14.
Use the properties of logarithms and the logarithms provided to rewrite each logarithm in
terms of the variables given.
15.
16.
15
13.
18.
Harder Problems
Condense each expression to a single logarithm.
16
Challenge
Summary/Closure:
Exit Ticket:
Homework: Page 331 – 332 #15 – 53 (every other odd)
17
18
Day 4 –Solving Logarithmic Equations
SWBAT: Solve Equations involving Logs
Warm - Up
19
Concept 1: Taking the log of both sides to solve an exponential equation
(Can’t change to the same base)
Solve for the variable in each problem.
If there is some reason the base is not isolated before you begin, ISOLATE the base first before you
take the log of both sides.
1.
2.
3.
4.
20
Concept 2: Equations with more than one log on one side.
If an equation has more than one “log,” use the rules for log distribution to CONDENSE the logs on a
side, and then proceed as you would otherwise. Usually this will involve converting the equation to
exponential form.
5.
6.
7.
8.

9.
If there are logs on each side of the equation, _______________________
10.
21
CHALLENGE
SUMMARY
Exit Ticket
Solve the equation. Round your answer to the nearest ten-thousandth.
Homework: Page 343 #3 – 14
Page 346 #3 – 17 (odd)
22
23
Day 5 –The Natural Log
SWBAT: solve equations using natural logs.
24
25
Rules for the Natural Log (Exactly the same as regular Logs)
Example:
26
Inverses
lnx and ex are inverse functions – If you do one, then the other, you get
what you started with. In symbols,
Ex. lnx = 4
ex = 12
27
Rules to solve an exponential equation with base e:
1. Isolate base e first.
2. Take the natural log of both sides to ”bring down the exponent”.
3. Solve for the variable.
Rules to solve an logarithmic equation with base e:
1. Condense each equation if possible to put all ln terms together
2. Exponeniate both sides to remove the ln function.
3. Solve for the variable.
31.
32.
28
33.

34.
Condense each equation if possible to put all log or ln terms together
If a Log or natural log is equal to another log of the same base then their contents are =.
35.
36.
Challenge
29
SUMMARY
Exit Ticket
Expand the natural logarithm.
30
Day 5 – HW
31
32
33
34
35
Day 7: Exponential Growth and Exponential Decay
SWBAT: Solve problems involving exponential growth, exponential decay
Warm-up:
Explain:
Exponential growth occurs when a quantity increases by the same rate r in each period t.
When this happens, the value of the quantity at any given time can be calculated as a function of
the rate and the original amount.
Exponential decay occurs when a quantity decreases by the same rate r in each time period t.
Just like exponential growth, the value of the quantity at any given time can be calculated by
using the rate and the original amount.
36
Concept 1: Finding “new” Value
Example 1:
The original value of a painting is $9,000 and the value increases by 7% each year. Write an exponential growth
function to model this situation. Then find the painting’s value in 15 years.
Step 1: Write the exponential growth function for this situation
y = _________ ( 1 ______ ) ____
y = _________ ( _____ ) ____
Step 2: Find the value in 15 years.
Example 2:
The population of a town is decreasing at a rate of 3% per year. In 2000 there were 1700 people. Write an
exponential decay function to model this situation, and then find the population in 2012.
Step 1 Write the exponential decay function for this situation
y = _________ ( 1 ______ ) ____
y = _________ ( _____ ) ____
Step 2 Find the value in 12 years.
Practice:
1) A sculpture is increasing in value at a rate of 8% per year, and its value in 2000 was $1200. Write an
exponential growth function to model this situation, and then find the sculpture’s value in 2006.
Step 1: Write the exponential growth function for this situation
y = _________ ( 1 ______ ) ____
y = _________ ( _____ ) ____
Step 2: Find the value in 2006.
Answer: ______________
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Example 3:
Is the equation A = 1500 (0.86)t a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1)
2)
3)
4)
exponential growth and 14%
exponential growth and 86%
exponential decay and 14%
exponential decay and 86%
Explain here!
Example 4:
Is the equation A = 5000 (1.04)t a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1)
2)
3)
4)
exponential growth and 4%
exponential growth and 96%
exponential decay and 4%
exponential decay and 96%
Explain here!
Example 5:
The fish population of Lake Collins is decreasing at a rate of 4% per year. In 2002 there were about 1,250 fish.
Determine whether this model is an exponential growth or exponential decay, and which equation can be used
to find the population in 2008?
1)
2)
3)
4)
exponential growth ;
exponential growth ;
exponential decay ;
exponential decay ;
y = 1250(0.96)6
y = 1250(1.04)6
y = 1250(1.04)6
y = 1250(0.96)6
Explain here!
Example 6:
The value of a gold coin picturing the head of the Roman Emperor Vespasian is increasing at the rate of 5 per
year. The coin is worth $105 now. Determine whether this model is an exponential growth or exponential
decay, and which equation can be used to find what the coin will be worth in 11 years?
1) exponential growth ;
2) exponential growth ;
3) exponential decay ;
y = 105(0.95)11
y = 105(1.05)11
y = 105(1.05)11
Explain here!
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Concept 2: Finding “Time”
Example 7:
The number of employees at a certain company is 1440 and is increasing at a rate of 1.5% per year. If the
number of employees now is 1646, how long did it take for the company to reach this many employees.
_______ = ______ ( 1 ______ )
1) The fish population in a local stream is decreasing at a rate of 3% per year. The original population was
48,000. If the number of fish now is 38,800, how long did it take for the fish to reach this population?
_______ = ______ ( 1 ______ )
2) The deer population of a game preserve is decreasing by 2% per year. The original population was 1850. If
the number of deer now is 1700, how long did it take for the deer to reach this population?
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Regents Questions:
40
Summary:
Exit Ticket:
1)
2) The value of a gold coin picturing the head of the Roman Emperor Marcus Aurelius is increasing at the rate
of 7 per year. If the coin is worth $145 now, what will it be worth in 14 years?
1) $308.44
2) $287.10
3) $373.89
4) $243.00
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Homework
Write an exponential growth function to model each situation. Then find the value of the function after
the given number of years.
1)
2)
3)
Write an exponential decay function to model each situation. Then find the value of the function after
the given number of years.
4)
5)
6)
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7) Is the equation A = 3200 (0.70)t a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1)
2)
3)
4)
exponential growth and 30%
exponential growth and 70%
exponential decay and 30%
exponential decay and 70%
Explain here!
8) Is the equation A = 1756 (1.17)t a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1)
2)
3)
4)
exponential growth and 17%
exponential growth and 83%
exponential decay and 17%
exponential decay and 83%
9) Is the equation A = 10,000 (0.45)t
Explain here!
a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1)
2)
3)
4)
exponential growth and 45%
exponential growth and 55%
exponential decay and 45%
exponential decay and 55%
10) Is the equation A = 5400 (1.07)t
Explain here!
a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1)
2)
3)
4)
exponential growth and 7%
exponential growth and 93%
exponential decay and 7%
exponential decay and 93%
Explain here!
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11)
12)
13)
\
44
Day 8 - Application of Exponential Equations
All of the following formulae, should they appear on my exam or the regents, will be given to
you. But you should know what they mean and how they are used. For ALL PROBLEMS,
round to the nearest TENTH when necessary.
Application #1: Exponential Growth and Decay
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Example 1: Astatine-218 has a half-life of 2 seconds. This means that every 2 seconds, half of the
radioactive material decays into non-radioactive material. Find the amount left from a 500 gram
sample of astatine-218 after 10 seconds.
Example 2: A bacterial culture triples every P hours. If the culture started with 13000 bacteria, and there are
24000 after 2 hours, what is the value of P in hours?
Example 3: A bacterial culture doubles every 5 hours. How long will it take for the culture to quadruple?
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You Try it!
1.
2.
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Application #2: Interest and Investments
Vocabulary:
 Principal is the initial investment (or loan) amount.
 Balance is the amount of money remaining on the loan or in your bank
 Interest is the money that is earned when someone lends someone money.
o Simple interest is interest calculated one time.
o Compound interest is interest calculated a certain number of times a year. The
interest percent quoted is SPLIT UP into a certain number of equal parts, and then it’s
compounded (calculated and paid out) that same number of times a year. You earn
interest on the interest. That’s why your money grows exponentially.
Final Value of Investment
(
)
A: amount (balance) after t years
P: Principal value
t: number of years that has passed
n: number of times per year interest is compounded
r: annual interest rate expressed as a decimal
Compounding…
Annually: ___ per year
Bi-Annually/Semi-annually:
___ per year
Quarterly: ___ per year
Monthly: ___ per year
Daily: ___ per year
Example 4: Write a compound interest function to model $1200 invested at a rate of 2%
compounded quarterly. Then find the balance after 3 years.
Example 5: Write a compound interest function to model $15,000 invested at a rate of 2.03%
compounded semi-annually. How long will it take for the amount of investment to triple?
48
You Try it!
1. Write a compound interest function to model $15,000 invested at a rate of 4.8% compounded
monthly. Then find the balance after 2 years.
2. Write a compound interest function to model $15,000 invested at a rate of 3% compounded
monthly. How long will it take for the amount of investment to reach $20,000?
49
Application #2, continued: Compounding Continuously, The number e and the
Natural Logarithm
There are two numbers for bases that are very important for applications of physics and mathematics.
One is base 10, and the other is base “ .”
is an irrational number, like . Because e is so important, there is also a special notation for
,
called “ln.”
is a couple of things:

as x gets very close to 0


as h gets very close to 0
(
) as n gets close to
You don’t need to know any of this
this year! It is very important in
finance and in calculus.
8
“Compounding Continuously” means that the interest on your (loan, borrowing, investment) is paid
out every teeny, tiny, unimaginably, infinitesimally small amount of time. It’s like compounding an
“infinite number of times.”
.
Investments Compounded Continuously
A = final amount of $ (balance)
P = Principal (Initial) investment
e = the irrational number that is approximately 2.718
r = rate expressed as a decimal
t= number of years of investment
Example 6: Write a compound interest function to model $1200 invested at a rate of 1%
compounded continuously. Then find the balance after 3 years.
Example 7: Write a compound interest function to model $15,000 invested at a rate of 3.05%
compounded continuously. How long will it take for the investment to reach $20,000?.
50
You Try it!
1. Write a compound interest function to model $15,000 invested at a rate of 2.8% compounded
continuously. Then find the balance after 2 years.
2. Write a compound interest function to model $15,000 invested at a rate of 1% compounded
continuously. How long will it take for the investment to reach $17,000?.
51
Application #3: The pH Scale
pH means “power of hydrogen.” It is the concentration of [H+] ions in a substance. For
example, HCL, a very strong acid, has a concentration of approximately 3.9 ×10−3. The pH of
HCL is about 3. Water, to which all other liquid substances is compared, is completely neutral,
with a concentration of 1×10−7 , has a pH of 7.
Example 8:
Determine
of a substanceofthat
has a concentration
of H+ of 2.34 x 10 -8
where
[H+] isthe
thepH
concentration
hydrogen
ions
Example 9: Determine the concentration of H+ for a substance that has a pH of 12.
You Try it!
1.
2.
52
SUMMARY
Exit Ticket
53
Day 9 – Graphs of Logarithmic Functions
Warm - Up:
Using the table below: a) Complete the table of values for y= 2x
b) sketch the graph of y= 2x
x
y
-2
-1
0
1
2
2) Recall:
How do we find the inverse of a function?
Properties of
Properties of
Domain:
Domain:
Range:
Range:
Asymptote:
Asymptote:
x-intercept:
x-intercept:
y-intercept:
y-intercept:
Find the inverse algebraically.
3) Graph the inverse of the function y = 2x.
54
55
56
57
58
59
Homework
1. Sketch below the graph of
the asymptote.
2. The expression
. Then, state the domain and range of the graph. Write the equation of
is equivalent to
1)
2) 2
3)
4)
3. Which expression is not equivalent to
?
1)
2)
3)
4)
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4. The expression
is equivalent to
1)
2)
3)
4)
5. The expression
1)
3)
2)
4)
6. Given:
is equivalent to
and
Express in terms of x and y:
7. Find the value of
8. If
correct to four decimal places.
, then the solution set for x is
1)
2)
3)
4)
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9. Drew’s parents invested $1,500 in an account such that the value of the investment doubles every seven
years. The value of the investment, V, is determined by the equation
, where t represents the
number of years since the money was deposited. How many years, to the nearest tenth of a year, will it take
the value of the investment to reach $1,000,000?
10. Solve for x:
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