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Name: ______________________________ Period: _______________________

College Bound Math Teacher: ___________________________

Exponents&

Logarithms

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Reference SheetAxis of symmetry: x=−b

2aQuadratic Formula:

x=−b±√b2−4 ac2a

Determinate of a 2×2 matrixA=|a b

c d|⇒ad−bc

Inverse of a 2×2 matrixA=[a b

c d ] A−1= 1

ad−cb [ d −b−c a ]

I = interest B = Balance P = principler = rate (as a decimal) n = number of compounding periods t = time in yearsExponential growth & decay

B = P(1 + r)t B = P(1 – r)t

Simple Interest

Compound Interest

Continuous Interest

Future Value – periodic

Present Value – one time

Present Value - periodic

I=prt B=P(1+ rn )nt

A=Pert B=P((1+ r

n )nt−1)

rn

P= B

(1+ rn )nt P=

B( rn )(1+ r

n )nt−1

Key Words:Simple interest

Key Words:Compounded

Annually, semiannually, quarterly, etc.

Key Words:Compounded continuously

Key Words:Total balance

Each/every

Key Words:Deposit now

Starting principal

goal

Key Words:Deposit now

Starting principalEach/every

goalTrigonometric Ratios

Sin A =

oppositehypotenuse Cos A =

adjacenthypotenuse Tan A =

oppositeadjacent

Coordinate Geometrym= Δy

Δx=y2− y1

x2−x1

Slope – Intercept Formula y = mx + b

Law of Sines asin A

= bsinB

= csinC

Law of Cosines (side) c2=a2+b2−2abCosC

(angle) CosC=a

2+b2−c2

2ab

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Area of a Triangle k=12abSinC

Lesson 1: Rules & Properties of ExponentsStudents will be able to apply the properties of exponents

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Lesson 2: Exponential Functions:

GROWTH & DECAYStudents will be able to apply the exponential growth and decay formulasStudents will be able to differentiate between using an exponential growth or decay.

*Many real world phenomena can be modeled by functions that describe how things grow or decay as time passes.  Examples of such phenomena include the studies of populations,

bacteria, the AIDS virus, radioactive substances, electricity, temperatures and credit payments, to mention a few.*

The exponential functions for growth and decay are represented as follows:

Exponential Growth Exponential Decay

Practice:

1. You deposit $1500 in an account that pays 6% interest compounded yearly. Find the balance after 5 years.

Equation:

Balance after 5 years:

2. You deposit $3500 in an account that pays 8.4% interest compounded yearly. Find the balance after 9 years.

Equation:

Balance after 9 years:

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3. You bought a Boston Whaler in 2004 for $12,500. The boat’s value depreciates by 7% a year. How much is the boat worth now? What will it be worth in 2020?

Equation: Equation:

Value this years: Value in 2020?

4. E. coli bacteria double in population every thirty minutes. If the initial population is 85, what’s the population of bacteria after three hours? After one day?

Equation: Equation:

Value this years: Value in 2020?

Classwork:1.) A population of Protozoa grows at a constant relative rate of .475 per day. On day zero, the population consists of two members. Find the population size after 10 days.

2.) The City of Schenectady currently has a population of approximately 66 thousand people. If the city lost approximately 2.8% of its population every year, in seven years, what would the new population be?

3.) Mendelevium-259 has a half-life of 6 hours. If you currently have 50 kg of Mendelevium-259, in a day how many kg of Mendelevium-259 would you still possess?

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One of the most common examples of exponential growth deals with bacteria. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling.  For example, if we start with only one bacteria which can double every hour, by the end of one day we will have over 16 million bacteria.

There is a well-known fable about a man from India who invented the game of chess, as a gift for his king.  The king was so pleased with the game that he

offered to grant the man any request within reason.  The man asked for one grain of wheat to be placed on the first square of the chess board, two grains to be placed on the second square, four on the third, eight on the fourth, etc., doubling the number of grains of wheat each time, until all 64 squares on the board had been used.  The king, thinking this to be a small request, agreed.  A chess board has 64 squares. How many grains of wheat did the king have to place on the 64th square of the chess board?Lesson #2: Homework1.) In 1995, there were 85 rabbits in Central Park.  The population increased by 12% each year.  How many rabbits were in Central Park in 2005?

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2.) A scientist has discovered a new strain of bacteria.  The bacteria culture initially contained 1000 bacteria and the bacteria are doubling every half hour. How many bacteria will there be after 2 and a half hours?

3.) At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate 0.02, what will be the population after 5 hours?

4.) Richard buys a car for $34,500 in 2005 and it depreciates on average at a rate of11% every year. What would be the estimated value of Richard’s car in 2012?

5.) A cup of coffee contains about 100 mg of caffeine. Every hour 16% of the amount of caffeine is metabolized and eliminated.a. Write an equation for C, the amount of caffeine in the body as a function

of t, the number of hours since the coffee was consumed.

b. How much caffeine is in the body after five hours, to the nearest hundredth of a milligram? After a day?

6.) The population of a particular species of bird increases at 2.5% annually. If there are 246 birds living in a region, how many will be there after 10 years?

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7.) From 1983 to 1997, the ratio of students per computer at a school has dropped by about 16.8% per year. If there were 103 students per computer in 1983, what was the number of students per computer in 1997?

8.) The rural town of San Filipposville has been losing population at a rate of 5.8% per year for the last 10 years. It has a current population of 12,500. What will the

population be in 8 years if it keeps declining at the same rate?

9.) Andy Graham has a savings certificate that is currently worth $10,500.00 and pays 6.5% interest compounded yearly. What is the balance when the certificate

matures in 5 years?

10.) Scientists, Matos and Massarelli, were working on an experiment which started with 10 bacteria. The bacteria doubled every hour for 24 hours. Write an

exponential model for this situation and use it to find the number of bacteria after 480 minutes.

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Lesson 3: Negative Exponents

But why?

Reason #1:

x3

x5 = subtract exponents =x−2= 1

x2x3

x5 = x⋅x⋅xx⋅x⋅x⋅x⋅x

= 1x⋅x

= 1x2

Reason #2:

2-3 2-2 2-1 20 21 22 23

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14

12 1 2 4 8

123

=18

122

= 14

121

=12

120

=11=1

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Lesson 4: Introduction to LogarithmsStudents will be able to perform inverses of exponents and logarithms.

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Students will be able to calculate the result from a logarithmic problem.

What is a Logarithm?In its simplest form, a logarithm answers the question:

How many of one number do we multiply to get another number?

Example: How many 2’s do we multiply to get 8?

So the logarithm is _________.

How to write it:

log2(8) = 3We are dealing with 3 numbers:

The base: The logarithm Desired outcomeThe number we are multiplying (a “2” in the example above).

How many times to multiply the base

The number you want to get out in the end

How do you say it? “The logarithm of 8 with a base of 2 is 3” Or “log base 2 of 8 is 3” Or “the base-2 log of 8 is 3”

More Examples:log5 (625) = ?

What does it mean?

Answer?

log2 (64) = ?

What does it mean?

Answer?

log3 (81) = ?

What does it mean?

Answer?

So logarithms answer a question like this:

2x=8

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The logarithm tells you what the exponent is!

The general formula is:

ax= y

log a( y )=x

Types of Logarithms

Common Logarithms: Base 10

Sometimes you will see a logarithm written without a base, like this:

Natural Logarithms: Base “e”

Another base that is often used is e (Euler’s Number) which is approximately 2.71828

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log(100)This means that the base is really 10.

It is called a “common logarithm”. Engineers love to use it.

It is how many times you need to use 10 in multiplication to get the desired number.

Example:

log (1000 )= log10(1000 )=3

This is called a “natural logarithm”. Mathematicians use this one a lot. On a calculator it is the “ln” button.

It is how many times you need to use “e” in multiplication to get a certain number.

Example:ln (7 . 389 )=loge (7 .389 )≈2

Because 2.718282 ¿ 2

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Negative Logarithms

Remember that negative exponents can occur, which means we can have negative logarithms as well!

A negative logarithm means how many times to divide by the number.

We could have just one division:Example:

What is log 8(0 . 125)= ???

Well,

18=0 . 125

so 8−1

= 0.125, so log 8(0 . 125)=−1

Or many divisions:Example:

What is log 2(0 .25 )= ???

Well,

14=0 .25= 1

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−2 = 0.25, so log 2(0 .25 )=−2

Number How Many 10s Base-10 Logarithm.. etc..        

1000 1 × 10 × 10 × log10(1000) = 3

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10100 1 × 10 × 10 log10(100) = 210 1 × 10 log10(10) = 11 1 log10(1) = 00.1 1 ÷ 10 log10(0.1) = -10.01 1 ÷ 10 ÷ 10 log10(0.01) = -2

0.001 1 ÷ 10 ÷ 10 ÷ 10 log10(0.001) = -3

.. etc..        

“Logarithm” is a word made up by Scottish mathematician John Napier (1550 – 1617), from the Geek word logos meaning “proportion, ratio or word” and arithmos meaning “number,” which together makes “ratio-number”!Logarithms Practice (Class work)

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To be able to evaluate logarithms that are not base 10 or the Natural log.

This will get you to the logBASE( feature that will allow you to perform any log.

Try log5 200

Guided Practice:

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9. You buy a new computer for $2100. The computer decreases by 50% annually. When will the computer have a value of $600?

10. You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by about 12%. How long until you have 10mg of caffeine?

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Lesson #4 Homework (Intro to Logarithms)

11. The foundation of your house has about 1,200 termites. The termites grow at a rate of about 2.4% per day. How long until the number of termites doubles?

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What careers use Logarithms?Answer via eHow.com

Logarithms, the inverses of exponential functions, are used in many occupations. Perhaps the most well-known use of logarithms is in the Richter scale, which determines the intensity and magnitude of earthquakes. Yet, there are many other professionals who use logarithms in their careers. Anyone who calculates the quantity of things that increase or decrease exponentially uses logarithms. This includes engineers, coroners, financiers, computer programmers, mathematicians, medical researchers, farmers, physicists and archaeologists. Because there is no definitive list of careers that require the use of logarithms, below is a brief sampling of how some careers employ these logs.

Coroner

You often see logarithms in action on television crime shows, according to Michael Breen of the American Mathematical Society. On such shows, coroners often attempt to determine how long a body has been dead. These television coroners, as well as their real-life counterparts, use logarithms to make such determinations. Once a body dies, it begins to cool. To calculate how long the body has been dead, the coroner must know how long the body temperature has not been at 98.6 degrees. Because the rate of the body cooling is proportionate to temperature differences between the body and its surroundings, the answer is found by calculating exponential decay using logarithms.

Medicine

Logarithms are used in both nuclear and internal medicine. For example, they are used for investigating pH concentrations, determining amounts of radioactive decay, as well as amounts of bacterial growth. Logarithms also are used in obstetrics. When a woman becomes pregnant, she produces a hormone known as human chorionic gonadotropin. Since the levels of this hormone increase exponentially, and at different rates with each woman, logarithms can be used to determine when pregnancy occurred and to predict fetus growth.

Actuarial Science

An actuary's job is to calculate costs and risks. Many of these calculations involve complicated statistics. For example, an actuary may work as a consultant designing pension plans for a company's employees. To do so, the actuary may have to figure out the chances of a particular 50-year-old employee living to be 89 years old. The actuary then designs that person's pension using statistics that are exponential in nature, and that's where the logarithms enter in.

Archaeology

Archaeologists use logarithms to determine the age of artifacts, such as bones and other fibers, up to 50,000 years old. When a plant or animal dies, the isotope of carbon, Carbon-14, decays into the atmosphere. Using logs, archaeologists can compare the decaying Carbon-14 to the Carbon-12, which remains constant in an organism even after death, to determine the age of the artifact. For example, this type of carbon dating was used to determine the age of the Dead Sea Scrolls.

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Lesson 5a: Expansion Properties of Logarithms

These rules are used to write a single complicated logarithm as several simpler logarithms (it’s called “expanding”). Notice that these rules work for any base.

1) logb(mn) = logb(m) + logb(n) (multiplication inside can be turned into addition outside)

2) logb(m/n) = logb(m) – logb(n) (division inside can be turned into subtraction outside)

3) logb(mn) = n · logb(m) (an exponent on everything inside can be moved out front)

Examples:Expand each expression (use expansion properties to expand as much as possible)

Problem Solution Which Rule?

log3 ( xy )

log7 ( xy )ln (x5 )

log ( x2

z 4 )ln ( xyz )

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Guided Practice

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Expand each logarithm – write down the rules you use as you expand (multiplication, division, and exponent).

Logarithmic Expansion Practice (Class work/ Homework)

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Lesson 5b: Condensing Properties of Logarithms

These rules are used to write several simple logarithms as a single complicated logarithm (it’s called “condensing”). Notice – these are the same rules as yesterday and that these rules work for any base.

1) logb(mn) = logb(m) + logb(n) (multiplication inside can be turned into addition outside)2) logb(m/n) = logb(m) – logb(n) (division inside can be turned into subtraction outside)3) logb(mn) = n · logb(m) (an exponent on everything inside can be moved out front)

Examples:Condense each expression (use contraction properties to simplify as much as possible)

Problem Solution Which Rule?

log 2( x )+log2 ( y )

log 9( x )− log9 ( y )

5 ln ( x )

ln (x )+ ln ( y )−2 ln( z )

2 log ( x )−3( log ( y )−log( z ) )

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Guided Practice

Condense each logarithm – write down the rules you use as you simplify (multiplication, division, and exponent).

Condensing Logarithms Practice (Class work/ Homework)

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Lesson 6: Basic Log Equations

Generally, there are two types of logarithmic equations. Study each case carefully before you start looking at the examples below.

The first type looks like this...

If you have a single logarithm on each side of the equation having the same base then you can set the arguments equal to each other and solve. The arguments here are the algebraic expressions represented by M and N.

 The second type looks like this...

If you have a single logarithm on one side of the equation then you can express it as an exponential equation and solve.

Examples:log 2(2x+8 )=log2(2 )

log5 ( x+6)=3

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log 4 (−3 x+1)=2 log7 (2 x+1 )= log7(4 x−9 )

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Solving Logarithmic Equations Practice (Class work/ Homework)

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Lesson 7: Applications of Logarithms: Growth and Decay

The following questions are from West Texas A&M’s (Agriculture and Manufacturing) College Algebra Course. (Feel free to check it out! http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut47_growth.htm)

Exponential Growth

A=A0ekt

A = the amount at the given time, tA0 = the initial amount before growth (t = 0)k = a constant growth rate written as a decimal k > 0 (must be POSITIVE to grow)t = the amount of time that has passede = Euler’s Number (think back to natural logs)

Example 1: The exponential growth model A=30e0 .019 tdescribes the

population of a city in the United States, in thousands, t years after 1994.  Use this model to solve the following:

A) What was the population of the city in 1994?

B) By what % is the population of the city increasing each year?

C) What will the population of the city be in 2005?

D) When will the city’s population be 60 thousand?

Example 2: A house is purchased for $150,000 in 2002. The value of the house is given by the exponential

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growth model A=150000 e0 . 0645t. Find when the house would be worth $200,000.

We also learned about exponential decay (remember the serial killer problem?). The formula for exponential decay looks the same, however your rate will now have to be NEGATIVE!!!

Exponential Decay

A=A0ekt

A = the amount at the given time, tA0 = the initial amount before growth (t = 0)k = a constant growth rate written as a decimal k < 0 (must be NEGATIVE to decay)t = the amount of time that has passede = Euler’s Number (think back to natural logs)

Example 3: An artifact originally had 12 grams of carbon-14 present. The decay model

A=12e−0 . 000121 tdescribes the amount of carbon-14 present after t years. How many grams

of carbon-14 will be present in this artifact after 10,000 years?

Example 4: A certain radioactive isotope element decays exponentially according to the model

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A=A0e−0 . 25t

, where A is the number of grams of the isotope at the end of t days and Ao is the number of grams present initially. What is the half-life of this isotope?

Example 5: Prehistoric cave paintings were discovered in a cave in Egypt. The paint contained 20% of the

original carbon-14. Using the exponential decay model for carbon-14, A=A0e

−0 . 000121 t,

estimate the age of the paintings.

Exponential Growth/Decay Practice: Class work/ Homework

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(Taken from Texas A&M College Algebra course – if you need help, go to the link at the top of the lesson – there is a step by step guide to the following problems)

The value of the property in a particular block follows a pattern of exponential growth. In the year 2001, your company purchased a piece of property in this block. The value of the property in thousands of dollars, t years after 2001 is given by the exponential growth

model V=500 e0 .055 t.

Use this model to solve the following:A) What did your company pay for the property?

B) By what percentage is the price of the property in this block increasing per year?

C) What will the property be worth in the year 2010?

D) When will the property be worth 750 thousand dollars?

1b. An artifact originally had 10 grams of carbon-14 present.  The decay model  

A=10e−0 .000121 t describes the amount of carbon-14 present after t years. 

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Use this model to solve the following:

A) How many grams of carbon-14 will be present in this artifact after 25,000 years?

B) What is the half-life of carbon-14?