Pre-AP Algebra 2

Unit 9 - Lesson 9 – Using a logarithmic scale to model the distance between planets and the Sun.

Objectives: Students will be able to read a graph with a logarithmic scale. Students will be able to use formulas for

decibels and magnitudes to determine and compare loudness and intensity of sound waves and earthquakes.

Materials: Hw #9-8 answers packet; Do Now, pair work, homework #9-9

Time Activity

5 min Homework Check Hand out answers packets to hw #9-8.

Pass around tally sheets.

10 min Homework Review

Present the solutions to the top 2-3 problems from the homework, based on the tally sheets.

10 min Do Now

Hand out the Do Now sheet with the planets data. Ask students to graph the distances of the planets on

the pair of axes. Rule: all data points must be included.

After students work on this for a few minutes, discuss some of the problems that they had completing the

task. The fact that the numbers are so far apart makes it difficult. If the scale is too small, then you can’t

get all the data points on the graph. If the scale is too big, many of the data points are squished so far

together that you can’t tell them apart.

We are going to use logarithms today to resolve these problems.

25 min Direct Instruction Ask students to turn to the back of the Do Now sheet. Label the second column “log d” and then

calculate the log of each distance. Once this is done, notice that the logs are much closer together.

Label the y-axis “log d”, create a good scale, and then make a bar graph.

Discuss the benefits of graphing with a log scale. Highlight the fact that each square on the y-axis now

represents a different amount, even thought the scale is constant. Moving up by a square is the same as

multiplying the underlying value by a factor of 10. Explain that these scales are used in several key real-

world applications: distance, magnitude of earthquakes, and loudness of sound. 30 min Pair Work

Hand out the Earthquakes handout to students to practice on. Review answers on the overhead.

Homework #9-9:

Pre-AP Algebra 2 Name: __________________________

Do Now

Logs… what are they good for?

The planets in the Milky Way galaxy orbit the sun at different distances1. Using the axes

given, scale the y-axis so that all the points given in the table can be plotted on the graph.

Remember – every square on the y-axis must be worth the same amount.

Planet Distance from

Sun

Mercury 36

Venus 68

Earth 92

Mars 141

Jupiter 484

Saturn 888

Uranus 1,783

Neptune 2,799

Pluto 3,67

1 Distances are in millions of miles so 36 means 36 million miles.

Follow-up questions.

1.) Milo is in orbit around the Sun at a distance of 630 million miles from the Sun.

What would the scaled value of this distance be?

Merc

ury

Ven

us

E

art

h

Mars

Jup

iter

S

atu

rn

Ura

nus

Ne

ptu

ne

Plu

to

M

erc

ury

V

enus

E

art

h

Mars

Ju

piter

Satu

rn

U

ranus

Ne

ptu

ne

P

luto

Planet

Mercury

Venus

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

Pluto

2.) Otis is in orbit around the Sun at a distance that has a scaled value of 4.9.

How many miles from the Sun is Otis?

For this question you need to divide Milo’s distance (in million miles) by Otis’s distance (in million miles). For

example, since Saturn is 888 million miles away from the Sun and Earth is 92 million miles away from the sun,

Saturn is 9.65 times further

888

92 9.65

3.) How many times is Milo further than Otis from the Sun?

Pre-AP Algebra 2 Name: __________________________

9-9 Pair Work

EARTHQUAKES!

Earthquakes can be measured using seismographs that can detect vibrations as little as 0.001mm in the ground.

These readings can be logarithmically converted into magnitudes and the Richter scale compares these

magnitudes. This is the function:

( ) log( ) 3M R R

where M is the magnitude and R is the seismic reading in 0.001 mm.

1. First finish the table

2. Make a bar graph on the axes below

Location M(R) R

San

Francisco

8

Tokyo 8.25

Chile 9

China 31,623

Indonesia 1,000,000

Loma

Prieta

6.9

El

Salvador

3,981

India 9.1

Los

Angeles

5,981

San F

rancis

co

1906

Tokyo

1923

Chile

1960

Chin

a

1976

Indo

nesia

2004

Lom

a P

rieta

1989

El S

alv

ador

2001

India

2001

Los A

nge

les

1994

1. In 1984 there was an earthquake in Mexico City that measured 8.1 on the Richter scale. Ten

years later there was an earthquake in Los Angeles that measured 6.6 on the Richter scale. In

order to accurately compare these two earthquakes do the following:

a. Find the seismographic reading for the earthquake in Mexico.

b. Find the seismographic reading for the earthquake in LA.

c. Using the seismographic readings, find how many times more intense the Mexico City

earthquake was than the Los Angeles.

2. Last weekend, a meteorite crashed to the Earth and registered a seismic reading of 398 mm.

Compare the intensity of the meteorite crash to that of the 8.2 magnitude earthquake that

happened in Chile in 1960 by finding the ratio of their seismographic readings.

Pre-AP Algebra 2 Name: _______________________

Homework #9-9

HW #9-9: Logarithmic Scales – LOUDNESS!

In class, you learned about how logarithms can be used to more easily plot out and work with large

distances and the intensity of earthquakes. Another application of logarithms is to deal with how

LOUD a sound is. To do this, we have developed a unit called decibels, which is a measure of the

loudness of a sound. This is the function:

L(x) = 10log(x) + 120

where L is the decibel level and x is the intensity measured in watts per square meter.

First calculate the decibel level for each sound. Then, plot the values on the graph, where the y-axis

represents decibel level.

Sound Intensity (measured

in watts/m2)

Decibel

Level

Softest noise

that can be

heard 10

-12

Whisper

across room 10

-11

Light rainfall 10-10

Normal

conversation 10

-7

Car driving,

heard from

50 ft away 10

-5

Noisy factory 0.001

Subway train 0.01

Rock concert 0.1

Shotgun blast 100

Soft

est no

ise th

at

can b

e h

eard

Wh

isper

acro

ss

room

Lig

ht ra

infa

ll

Norm

al

convers

atio

n

Car

drivin

g,

heard

from

50 ft

aw

ay

Nois

y facto

ry

Subw

ay tra

in

Rock c

oncert

Shotg

un

bla

st

You have just practiced finding decibel levels given the intensity. Now, use the same function (but

solving in the other direction) to find the intensity of the following sounds:

Hockey Crowd: 120 decibels Intensity:

Heavy City Traffic: 90 decibels Intensity:

Dripping Faucet: 25 decibels Intensity:

Threshold for Human Pain: 130 decibels Intensity:

Now, let’s compare intensities of sounds. Suppose that Kade said “I love logarithms!”, and his voice

was measured at 40 decibels. Then, Lorenzo shouted, “People, I finished my homework!”, and his

voice was measured to be 80 decibels. How many times more intense was Lorenzo’s voice?

To do this, first find the intensity of each sound, just like you were doing above.

Kade’s intensity:

Lorenzo’s intensity:

Now, find the ratio of Lorenzo’s intensity to Kade’s intensity and simplify:

Notice that Kade was twice as loud (80 is twice as loud as 40), but his intensity was _____ times

greater!

And now, on to the review packet! Next class is on May 2nd

(B-Day) or May 3rd

(A-Day). You will

be finishing your projects. The unit test will be May 6th

(B-Day) or May 7th

(A-Day).

Pre-AP Algebra 2 Name: _______________________

Review Packet

The Big Logarithm Review Packet!

Unit test is coming next week – do a couple pages a

day!

Part 1: Rational Exponents

The definitions: a1/ n a

n

am/ n a

n m

Evaluate each expression. Converting back and forth between radical and exponential form is

helpful. Changing decimals to fraction form is also helpful.

1) (125)2/3 2) 493/2 3) (27)5/3 4) (64)1/3

5) 1003/2 6) 216 4 /3

7) 643/2 8) 25

121

3/2

9) 8

81

2/3

10) 361.5 11) 1442.5 12) 1

9

1.5

13) 5 6

14) 174 8

15) 99 2

16) 113 6

17) 3.24 4

18) 12345 5

Simplify each expression by using the properties of

exponents.

19) 85/3 81/3 20)

x2/3

1/ 2

=

21)

53/ 4

51/ 2 22)

121/5

123/5

23) 351/5 353/5

5/4

24)

32

72

1/ 2

Name Property

Product of Powers am an amn

Power of a Power

am

n

amn

Power of a Product

ab

n

an bn

Quotient of Powers

am

an amn

Power of a Quotient

a

b

n

an

bn

Part 2: Graphing Exponential Functions

Determine if each function will show exponential growth or decay. Also determine the y-intercept.

Then make a table of values and graph each function on the coordinate plane.

1)

y 21

2

x

2) y

2

33

x

3) y 4(2)x 4) y 2(1.5)x

Part 3: Percents

Convert each percent to a fraction, and reduce.

1) 5% 2) 24% 3) 140% 4) 22.5% 5) 800%

Find each percent.

6) 82% of 154 7) 15.4% of 244 8) 2.5% of 15

9) 100% of 256.5 10) 200% of 350 11) 1000% of 75

Answer each question, thinking about the best way to calculate percent increase/decrease.

12) At a restaurant, you are supposed to leave a tip that is at least 15% of the bill. Your bill was $45.

How much do you leave in total?

13) Your beautiful new car was worth $18,000, until the side got dented in. Now, it has lost 23% of

its original value. How much is the car worth?

14) You invested $6000 into your savings account. A few years later, the amount grew to $9500.

What percent did your savings increase by?

15) I really like to eat tofu, so I buy it in bulk. I had a 12-pound box that I bought a few weeks ago; I

just weighed it again, and now there are only 7.5 pounds left. What percent did my total amount

of tofu decrease by?

Part 4: Exponential Modeling and the Number e

For each situation, determine a) the initial value; b) if it is growth or decay; c) the percent

increase/decrease.

1) The value of a car over time is modeled by the function v(t) = 35,000(0.66)t, where v is in

dollars and t is in years.

2) The value of my sister’s house in San Jose is modeled by v(t) = 450,000(1.09)t, where v is in

dollars and t is in years.

3) A waste product of producing nuclear energy is radioactive plutonium. A given quantity of

plutonium is stored in a container. The amount P (in grams) of plutonium present in the

container after t years can be modeled by P = 1000(0.993)t.

For each problem, write a function that models the situation. Then, answer the questions.

1) The petting zoo director purchased 10 guinea pigs. Each month, the number of guinea pigs

increase by 85%.

a. Function:

b. How many guinea pigs will there be after 8 months?

c. How long will it take for there to be 100,000 guinea pigs? (That’s a lot!)

2) When you drop a sugar cube into a glass of hot water, the amount of sugar remaining in the

cube decreases by 10% per second. Assume that a sugar cube weighs 3 grams.

a. Function:

b. How many grams will there be left after 10 seconds?

c. How long will it take for there to be only ½ gram left?

3) You put $3000 into a savings account that has continuously compounded interest, at an

annual rate of 7.5%.

a. Function:

b. How much money will you have after 3 years?

c. How long will it take for your money to triple?

4) A scientist has recently discovered 500 grams of radioactive material at a local park. Each

week, she measures less material from the week before and she is able to come up with the

following model for the decay: A(t) = 500e-.035t

where A is the total grams of radioactive

material and t is the number of weeks.

a. How many grams will be left after 4 weeks?

b. How long will it take for there to be only ¼ of the original amount left?

Part 5: Estimating and Evaluating Logarithms

To convert between log and exponential form: log

bx a ba x

Estimate the value of each logarithm (find two consecutive integers that it falls between) without

using your calculator.

1) log325 2) log250 3) log 5000 4) log1/2(1/10)

Evaluate each logarithm without using your calculator.

5) log381 6) log1/416 7) log7(1/343) 8)log1/3(1/81)

9) log51 10) ln e 11) log1717 12) log6.56.5

Find the value of x in each log expression.

13) logx32 = 5 14) log3x = -6 15) log1/28 = x 16) logx7 = ½

17) log x = 7 18) ln e5 = x 19) log .01 = x 20) log8x = 1/3

Change each logarithm to base 10 and then use your calculator’s LOG button to estimate its value.

21) log382 22) log7.159 23) log2/3(9/11)

Change each logarithm to base e and then use your calculator’s LN button to estimate its value.

24) log382 25) log7.159 26) log2/3(9/11)

Solve each equation by isolating the power, then converting to a logarithm. Write the exact answer

(as a logarithm), and then use the Change of Base Theorem and your calculator to estimate the

solution to the thousandths place.

27) 7x = 256 Exact: Estimate:

28) 5.2(3.1x) = 146.3 Exact: Estimate:

29) -2(2.5x) + 39 = 3 Exact: Estimate:

Part 6: Properties of Logarithms

Special Cases (Matching Bases): blog

bM

and log

bbM both equal M.

The three properties:

Multiplication: logb

AC logb

A logbC

Division: logb

A

C

log

bA log

bC

Power: logb

AC C logb

A

Use log

tA 0.4 and

log

tB 2 to find the exact value of each expression:

1) log

tAB 2)

log

t

B

A 3)

log

tA3

4) log

tA3B 5)

log

tAB

2

6) log

t

B

A

Expand each expression using the properties of logs.

7) log

28x3 8)

log

7x

5 9)

log7

49x5

y2

10) ln x

4 2

11) log 2x3 y

Condense each expression using the properties of logs.

12) log

53 log

59 13) 2log x 3log2 14)

1

2ln x 3ln5

15) 2 log

58 log

52

1

2log

53 16)

2log x 3log5

2

3log27

17) log

5(x 1) log

5(x 3) 18)

log 2x 5x2 3log x

Part 7: Solving Logarithmic and Exponential Equations

Solve each equation. Make sure to check your answers and cross out any extraneous solutions.

1) 4 log5 x log51

81 2)

1

2log5 x 2log5 3

3) log2(x 1) log2(x 2) 2 4) log x log(x 5) 2

For 5 and 6, solve by isolating the power and then taking log10 of both sides.

5) 2(3.5)x 28 4 6) 2

3(0.25)x 8 14

7) 2x5 314x 8) 10x2 2x1

HW 9-8 Answer Key

1) x = 38

2) x = ±2

3) x = 11

4) x = 4 or x = -1

5) x = 14

6) x = (ln4)/1.07 ≈ 1.296

7) 445.6

4

1log 24.1 x

8) 8 53x

9) 1,2 xx

10) 77.3

3

64log x

11) 044.217log4 , same

HW 9-8 Tally Sheet

1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

11)

HW 9-8 Tally Sheet

1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

11)