Chapter 1 Section 1.1 Scientific Notation Powers of Ten

0001.10

001.10

01.10

1.10

110

1010

10010

100010

1000010

4

3

2

1

0

1

2

3

4

Standard Scientific Notation

10110 NwherexN n and n is an integers Examples of numbers in scientific notation

11

8

1017.4

104.3

x

x

Using Scientific Notation The population of Mexico City is about 23,000,000 To change the number into scientific notation you move the decimal place seven places to get: 7103.2 x The speed the speed of light is 30,000,000,000 m/s. Write this number in scientific notation. Answer: 10100.3 x

Example 1 Convert .00000079 to scientific notation Answer: 8109.7 x

Example 2 Convert .000000000043 to scientific notation Answer: 12103.4 x Example 3 Convert 8101.5 x to decimal notation. Answer: 000,000,510 Example 4 Convert 51011.3 x to decimal notation. Answer: 311,000 Using operations with scientific notation Multiplication with scientific notation Example 5 Simplify )108.7)(101.6( 76 xx

14

13

76

76

10758.4

1058.47

10)8.7)(1.6(

)108.7)(101.6(

x

x

x

xx

Example 6 Simplify )107)(103( 710 xx

18

17

710

710

101.2

1021

10)7)(3(

)107)(103(

x

x

x

xx

Example 7 Simplify )104)(105( 610 xx

5

4

610

610

100.2

1020

10)4)(5(

)104)(105(

x

x

x

xx

Division with scientific notation Example 8

Simplify 8

12

101.2

102.4

x

x

4

812

8

12

102

101.2

2.4

101.2

102.4

x

x

x

x

Example 9

Simplify 3

9

105.3

108.5

x

x

6

39

3

9

1066.1

104.3

8.5

105.3

108.5

x

x

x

x

Example 10

Simplify 8

410

102

)104.3)(102.4(

x

xx

6

8

14

8

410

1019.7

102

1038.14

102

)104.3)(102.4(

x

x

x

x

xx

Example 11

Simplify 10

85

103

)102.1)(102(

x

xx

2

3

10

13

10

85

100.8

108.

103

104.2

103

)102.1)(102(

x

x

x

x

x

xx

Example 12 The National debt is about 12108.7 x , and there about 296,000,000 Americans. What would be the debt per American citizen? Convert the population of America to scientific notation. 81096.2000,000,296 x

Each American would owe: 350,26$10635.2$1096.2

108.7$ 48

12

xx

x

Estimation Example 13 You make $13.85 per hour, about how much would make in a year assuming you work 40 hours a week? Round $13.85 to $14.00 Salary per week: 560$00.14$40 x Salary per year: 100,29$560$52 x Example 14 A high graduate makes about an average of $25,000 per year while a college graduate makes about an average of $40,000 per year.

000,600,1$000,40$40:

000,000,1$000,25$40:

xGraduateCollege

xGraduateSchoolHigh

Section 1.2 Percents Introduction to Basic Percents The word percent translates to mean “out of one hundred”. A score of 85% on test means that you scored 85 points out of 100 possible points on the test. If you scored 44 out of 50 points on a test, then this would be a percent value of 88%. This value can be obtained by multiplying the numerator and denominator by 2 as shown in the next illustration.

%8888.100

88

)50(2

)44(2

50

44

Since a percent represents the amount out of a hundred, to change a percent to a decimal, you simply drop the percent symbol and divide by 100 which can be done by moving the decimal two placing to the left as shown in the next examples.

645.100

5.64%5.64

45.100

45%45

Basic Percent Problems One of the basic uses of percents is to find the percent amount of a given number. For example, how you would take 34% of 60? This would be done by changing 34% to .34, and then multiplying by .34 by 60 as shown here:

4.20)60)(34(.34.%34 Example 1 What is 46% of 90?

4.41)90)(46(.

46.%46

Mark up, mark down, and sales price There are many common uses for percents in our society. As consumers, people use percentages to find sales prices, mark up prices, and discount. In this section, we will study how to use percents to compute discounts, mark up prices, sales prices, and sales tax. The first of these topics we will explore are discount and sales price. Discount Discount = (Percent Mark Down)(Retail Price) Sale Price Sale Price = Retail Price – Discount Example 2 A men’s sports jacket that has a retail price of $170 is discounted by 25%. What is the sale’s price of the sports jacket?

50.127$50.42$170$

50.42$170$25.

25.%25

priceSales

Discount

DownMark

Example 3 A pair of jeans that has a retail price of $55.00 is discounted at 30%. What is the sale’s price of the jeans?

50.38$50.16$55$

50.16$00.55$30.

30.%30

priceSales

Discount

DownMark

Example 4 The sale price of a VCR is $110.00. If the mark down is 30%, find the retail price of the VCR.

14.157$70.

00.110

70.

70.

00.11070.

00.11030.

00.110$

30.

x

x

x

xx

pricediscount

discountx

priceoriginalxLet

Mark Up Price When stores purchase items at a whole sale price, the retail price is computed by marking up the whole sale cost using the given formulas. Mark Up = (Percent Mark Up)(Whole Sale Price) Retail Price = Whole Sale Price + Mark Up Example 5 A store purchases DVD players at a whole sale price of $30 per unit which is to be marked up by 80%. What will be the retail price of the DVD player?

00.54$00.24$00.30$

00.24$)00.30)($80(.

80.%80

priceretail

upmark

upmarkpercent

Example 6 The whole sale price of a pair of jeans is $20.00. If the jeans are marked up by 65%, what is the retail price of the jeans?

00.33$00.13$00.20$

00.13$)00.20)($65(.

65.%65

priceretail

upmark

upmarkpercent

Example 7 The retail price of a new television that has been marked up by 75% is $300.00. Find the whole sale price of the television.

43.171$75.1

00.300

75.1

75.1

00.30075.1

00.30075.

00.300$

75.

x

x

x

xx

pricediscount

discountx

priceoriginalxLet

Sales Tax When items are purchased at a store or place of business, a state sale’s taxes is calculated and added on the price of the item. The percent rate of sale’s tax in the United States is determined by each state. For example the sales tax in Virginia is 4.5%. Some states such as Delaware and Montana do not have any sale’s tax. The state sale’s tax is calculated by multiplying the percent rate by the purchase price. The state sale’s tax is then added on the purchase price of the item. Sales Tax Formula Sale’s Tax = (sale’s tax rate)(purchase price)

Example 8 The state sale’s tax rate in Virginia is 4.5%. Find the full cost to purchase a $50 pair of shoes using the Virginia tax rate of 4.5%.

25.52$25.2$00.50$

25.2$)50)(045(.

taxincludingCost

taxsales

Example 9 The state sale’s tax rate in Ohio is 6%. Find the full cost to purchase the same pair of shoes in problem 7 using the Ohio tax rate of 6%.

00.53$00.3$00.50$

00.3$)50)(06(.

taxincludingCost

taxsales

Problem Set (Section 1.2)

1) Find the discount on each item if the mark down rate is 5%. a) $90.00 b) $25.00 c) $130.00

2) Find the discount on each item if the mark down rate is 15%.

a) $100.00 b) $45.00 c) $140.00

3) Find the sale’s price on each item given the mark down rate is 20%. a) $120.00 b) $400.00 c) $215.00 4) Find the sale’s price on each item given the mark down rate is 15%. a) $60.00 b) $130.00 c) $15.00 5) A pair of jeans that has a retail price of $42.00 is discounted at 25%. What is the sale’s price of the jeans?

6) A women’s dress that has a retail price of $80 is discounted by 35%. What is the sale’s price of the dress? 7) The sale price of a television is 200.00. If the mark down is 22%, find the retail price of the television. 8) The sale price of a laptop computer is $1100.00. If the mark down is 10%, find the retail price of the laptop computer. 9) Using a mark up rate of 30%, find the retail price given the whole sale price of each item. a) $140.00 b) $30.00 c) $75.00 10) Using a mark up rate of 45%, find the retail price given the whole sale price of each item. a) $200.00 b) $34.00 c) $124.00 11) The wholesale price of a pair of dress pants is $25.00. If the jeans are marked up by 60%, what is the retail price of the pants? 12) The wholesale price of a CD player is $57.00. If the CD player is marked up by 30%, what is the retail price of the CD player?

13) The retail price of a pair of dress pants is $70.00. If the jeans are marked up by 25%, what is the whole sale price of the pants? 14) The retail price of a new television that has been mark up by 55% is $420.00. Find the whole sale price of the television. 15) The sale’s tax rate in North Carolina is 4.5%. Find the total cost including sale’s tax for purchasing each item. a) $150.00 b) $340.00 16) The sale’s tax rate in Michigan is 6%. Find the total cost including sale’s tax for purchasing each item. a) $250.00 b) $420.00

Section 1.3 Introduction to Mathematical Modeling Types of Modeling 1) Linear Modeling 2) Quadratic Modeling 3) Exponential Modeling 4) Logarithmic Modeling Each type of modeling in mathematics is determined by the graph of equation for each model. In the next examples, there is a sample graph of each type of modeling Linear models are described by the following general graph

Quadratic models are described by the following general graph

Exponential models are described by the following general graph

Logarithmic Models are described by the following general graph.

Section 1.4 Linear Models Before you can study linear models, you must understand so basic concepts in Algebra. One of the main algebra concepts used in linear models is the slope-intercept equation of a line. The slope intercept equation is usually expressed as follows: Standard linear model

Interceptyb

slopem

bmxy

In this equation the variable m represents the slope of the equation and the variable b represents the y-intercept of the line. When studying linear models, you must understand the concept of slope. Slope is usually defined as “rise over run” or “change in y over change in x”. In general slope measures the rate in change. Thus, the idea of slope has many applications in mathematics including velocity, temperature change, pay rates, cost rates, and several other rates of change. Slope

xinchange

yinchange

run

riseSlope

12

12

xx

yym

Basic Algebra Skills (Slope and y-intercept) In next examples, we will find the slope of a line given two points on the line. Example 1 Find the slope between the points (1,3) and (3,2)

2

1

2

1

13

32

12

12

xx

yym

Example 2 Find the slope between the points (2,3) and (4,6)

2

3

24

36

12

12

xx

yym

Slope and y-intercept also can be found from the equation in slope-intercept, as shown in this next example. Notice that the equation is written in slope-intercept form. Example 3 Find the slope and y-intercept

2

3

23

b

m

xy

If the equation is not written in slope intercept form, it can be rearranged to slope-intercept form by solving the equation for y. This procedure is shown in the next two examples. Example 4 Find the slope and y-intercept

23

2

23

23

6

3

2

3

3

623

62322

632

b

m

xy

xy

xy

xyxx

yx

Example 5 Find the slope and y-intercept

25

3

25

35

10

5

3

5

5

1035

103533

1053

b

m

xy

xy

xy

xyxx

yx

Example 6

Graph the equation 22

3 xy

First construct a table using 4 arbitrary values of x, and then substitute these x values to

the equation 22

3 xy to get the corresponding y values.

x 2

2

3 xy

1

2

12

2

32)1(

2

3y

2 1232)2(

2

3y

3

2

52

2

92)3(

2

3y

4 4262)4(

2

3y

Next make point using the four points in the above table.

4

2

-2

-4

-6

-5 5

Applications of Linear Equations Example 6 (Temperature conversion)

325

9 CF

a) Sketch a graph of 325

9 CF

C

325

9 CF

10 50321832)2(932)10(

5

9F

20 68323632)4(932)20(

5

9F

30 86325432)6(932)30(

5

9F

40 104327232)8(932)40(

5

9F

b) Use the model to convert 120 degrees Celsius to degrees Fahrenheit.

248

32216

32)120(5

9

325

9

F

F

F

CF

c) Use the model to convert 212 degrees Fahrenheit to Celsius.

CC

C

C

C

C

CF

0100

5

9

9

5)180(

9

55

9180

32325

932212

325

9212

325

9

Example 7 (Business Applications) The revenue of a company that makes backpacks is given by the formula xR 50.21 where x represents the number of backpacks sold.

a) Graph the linear model xR 50.21 X xR 50.21 10 215)10(50.21 R 20 430)20(50.21 R 30 645)30(50.21 R 40 860)40(50.21 R

b) Use the model to calculate the revenue for selling 50 backpacks

0.1075$)50(5.2150.21

50

xR

x

c) What is the slope

50.21$m d) What is the meaning of the slope?

Cost per unit sold Revenue made per backpack solid

Example 8 (Sales) A salesperson is paid $100 plus $60 per sale each week. The model 10060 xS is used to calculate the salesperson’s weekly salary where x is the number of sales per week. a) Graph 10060 xS x 10060 xS 2 220100120100)2(60 S 4 340100240100)4(60 S6 460100360100)6(60 S8 580100480100)8(60 S

b) Use the model to calculate the salespersons weekly salary if he/she makes 8 sales.

00.580$100480100)8(60 S

c) What is the slope of the equation

salem

$60

d) What is the meaning of the slope

Dollars per each sale

Example 9 Given the following data sketch a graph Time Temperature 1 min C03 2 min C07 3 min C011 4 min C014 Sketch a graph of the given data and then compute the slope of the resulting line.

12

10

8

6

4

2

-2

-5 5 10 15

(2,7)

(1,3)

Use the points (1,3) and (2,7) in the above graph to compute the slope

41

4

12

37

m

Example 10 An approximate linear model that gives the remaining distance, in miles, a plane must travel from Los Angeles to Paris given by td 5506000 where d is the remaining distance and t is the hours after the flight begins. Find the remaining distance to Paris after 3 hours and 5 hours.

milesd

d

d

4350

16506000

)3(5506000

milesd

d

d

3250

27506000

)5(5506000

How long should it take for the plane to flight from Los Angeles to Paris?

hourst

t

t

ttt

t

9.10550

6000

550

550

6000550

55055060005500

55060000

Problem Set 1.4 1) Find the slope between the points (1,1) and (3,5) 2) Find the slope between the points (0,0) and (4,5) Given the equation, find the slope and y-intercept.

3) 24

3 xy

4) 643 yx 5) 632 yx Graph the following equations 6) xy 3 7) 5 xy

8) 14

1 xy

9) xy 6 Linear Models 10) The revenue of a company that makes backpacks is given by the formula xR 50.34 where x represents the number of backpacks sold.

a) Graph the linear model xR 50.34 b) Use the model to calculate the revenue for selling 40 backpacks? c) What is the slope of the model? d) What is the meaning of the slope?

11) A salesperson is paid $100 plus $30 per sale each week. The model 10030 xS is used to calculate the salesperson’s weekly salary where x is the number of sales per week.

a) Graph 10030 xS b) Use the model to calculate the salespersons weekly salary if he/she makes 8 sales. c) What is the slope of the equation? d) What is the meaning of the slope?

12) A salesperson is paid $200 plus $50 per sale each week. The model 20050 xS is used to calculate the salesperson’s weekly salary where x is the number of sales per week.

a) Graph 20050 xS b) Use the model to calculate the salespersons weekly salary if he/she makes 8 sales. c) What is the slope of the equation? d) What is the meaning of the slope?

13) An approximate linear model that gives the remaining distance, in miles, a plane must travel from San Francisco to London given by ttd 5005500)( where )(td is the remaining distance and t is the hours after the flight begins. Find the remaining distance to London after 2 hours and 4 hours.

Section 1.5 Quadratic Models Graph of Quadratic Models

The graph of a quadratic model always results in a parabola. The general form of a quadratic function is given in the following definition. A quadratic function is a function where the graph is a parabola and the equation is of the form: cbxaxy 2 where 0a

The x-coordinate of vertex is given by the equation: a

bx

2

The vertex is the turning point on the graph of a parabola. If the parabola opens upward, then the vertex is the lowest point of the graph. If the parabola opens downward, then the vertex is the highest point on the graph. The direction of the parabola opens can be determined by the sign of the “ 2x ” term or the a term in the above equation. If 0a , then the parabola open downward. Similarly if 0a , then the parabola opens upward. (See graphs below in figure 1-1) Figure 1-1 A parabola where 0a and the vertex is the lowest point on the graph

A parabola where 0a and the vertex is the highest point on the graph

Here are some examples of finding the vertex and x-intercepts of an exponential equation. The graph of the quadratic equation is also provided in these examples Example 1 Find the vertex and x-intercepts of the quadratic equation, and then make a sketch of the parabola.

02

0

)1(2

0

3,1

32

x

ca

xy

x-intercepts:

)0,3()0,3(

3

3

3

03

2

2

2

and

x

x

x

x

Graph for Example 1

Example 2 Find the vertex and x-intercepts of the quadratic equation, and then make a sketch of the parabola.

4

9

2

9

4

9

2

33

2

3

2

3

)1(2

3

3

2

2

y

x

Vertex

xxy

x-intercepts

)0,3()0,0(

3

030

030

0)3(

032

and

x

xx

xorx

xx

xx

Graph of the function

Example 3 Find the vertex and x-intercepts of the quadratic equation, and then make a sketch of the parabola.

)3,1(

3631613

16

6

)3(2

)6(

63

2

2

y

x

Vertex

xxy

x-intercepts

)0,2()0,0(

2

020

0203

0)2(3

063 2

and

x

xx

xorx

xx

xx

Graph of 063 2 xx

More about Quadratic Equations In some instances, the quadratic equation will not factor properly. In this case, you must use what is called the quadratic formula. In the next few examples, the quadratic formula will be used to find the solutions of a quadratic equation. The Quadratic Formula The solution to the equation cbxaxy 2 is given by

a

acbbx

2

42

Example 4 Solve 0752 xx

2

535

2

28255

)1(2

)7)(1(455

2

4

7

5

1

22

a

acbbx

c

b

a

Example 5 Solve 0972 xx

14

857

)7(2

36497

)7(2

)9)(1(477 2

x

Example 6 At a local frog jumping contest. Rivet’s jump can be approximated by the equation

xxy 26

1 2 and Croak’s jump can be approximate by xxy 42

1 2 , where x = the

length of jump in feet and y = the height of the jump in feet. a) Which frog can jump higher

Rivet’s vertex: 6

3

12

6

12

2

x Height: fty 6126)6(2)6(6

1 2

Croak’s vertex: 41

4

2

12

4

x Height: fty 8168)4(4)4(2

1 2

Croak can jump higher at 8 feet b) Which frog can jump farther Rivet’s can jump farther at 2(6 ft) = 12 feet

Graph of the frogs jumps

8

6

4

2

-2

-5 5

g x = -1

2 x2+4x

f x = -1

6 x2+2x

Using the parabola to find the maximum or minimum value of a quadratic function The parabola can be used to find either the maximum value or the minimum value of a quadratic function. (See figure 1-1) This can simply be done by find the vertex of the parabola. Remember as stated earlier the vertex will turn out to be either the highest point on the curve or the lowest point on the curve. In the next examples, the vertex of the parabola will be use to find the maximum value. Example 7 The path of a ball thrown by a boy is given by the equation xxy 5.104. 2 where x is the horizontal distance the ball travels and y is the height of the ball. Find the maximum height of the ball in yards. Find the vertex of the ball

yardsy

x

141.281.14)75.18(5.175.1804.

75.1808.

5.1

)04.(2

5.1

2

Example 8 The path of a cannon ball is given by the equation xxy 0.61. 2 where x is the horizontal distance the ball travels and y is the height of the cannon ball. Find the maximum height of the cannon ball in feet.

Find the vertex of the cannon ball.

feetyx 9018090)30(6301.302.

0.6

)1.(2

0.6 2

Problem Set 15 Find the vertex and x-intercepts of the given parabola, and then make a sketch of the parabola. 1) xxy 42 2

2) 42 xy

3) 122 xxy

4) 342 xxy

5) 162 xy

6) xxy 63 2 Quadratic Models 7) The path of a ball thrown by a baseball player is given by the equation

xxy 6.102. 2 where x is the horizontal distance the ball travels and y is the height of the ball. Find the maximum height of the ball in yards. 8) The path of a ball thrown by a boy is given by the equation xxy 8.106. 2 where x is the horizontal distance the ball travels and y is the height of the ball. Find the maximum height of the ball in yards. 9) The path of a cannon ball is given by the equation xxy 0.605. 2 where x is the horizontal distance the ball travels and y is the height of the cannon ball. Find the maximum height of the cannon ball in feet. 10) The path of a cannon ball is given by the equation xxy 0.81. 2 where x is the horizontal distance the ball travels and y is the height of the cannon ball. Find the maximum height of the cannon ball in feet.

Section 1.6 Exponential models The exponential function

718.2e “The Euler number” Example 1: Simplify the following exponential functions

40.1)3

05.1

)2

39.7)1

3

1

33

2

e

ee

e

The graph of the exponential function Example 2 Graph xey x xey -2 14.2 ey -1 37.1 ey 0 1 oey 1 7.21 ey 2 4.72 ey

Example 3 Graph xey 2.10 x Y -2 7.61010 4.)2(2. eey -1 2.81010 2.)1(2. eey 0 101010 0)0(2. eey 1 2.121010 2.)1(2. eey 2 9.141010 4.)2(2. eey

Exponential Models Exponential models are used to predict human populations, animal populations, money growth, pollution growth, and other aspects of society that fit exponential models. The variable of an exponential model is found in the exponent of the equation.

Exponential Growth

timet

rater

ValueOriginalP

ValueNewP

rPP t

0

0 )1(

Example 4 The population of the United States is 290 million, what would be the population of the U. S. be in 20 years if its population would growth at a steady rate of .7 % for 20 years?

333416746)007.1(290000000)007.1(290000000

20

007.%7.

000,000,290

)1(

2020

0

0

P

t

r

P

rPP t

Example 5 The population of Blacksburg, Virginia is 41,000, what would be the population in 10 years if Blacksburg would grow at a rate of 1.1 % per year?

45740)011.1(41000)011.1(41000

10

011.%1.1

41000

)1(

1010

0

0

P

t

r

P

rPP t

Example 6 In 1995 the United States had greenhouse emissions of about 1400 million tons, where as China had greenhouse emissions of about 850 million tons. If in the next 25 years China greenhouse emission grew by 4 percent and the U. S. greenhouse emission grew by 1.3 percent, what would the emissions in tons for both countries in 2020?

tonsmillionP

t

r

millionP

rPP

inEmissionsSUt

1933)013.1(1400)013.1(1400

25

013.%3.1

1400

)1(

2020..

2525

0

0

tonsmillionP

t

r

millionP

rPP

inEmissionssChinat

2265)04.1(850)04.1(850

25

04.%0.4

850

)1(

2020'

2525

0

0

Example 7 Using the exponential growth formula, find the amount of money that you would have in a bank account if you deposited $3,000 in the account for 15 years at 1.1 % interest rate?

91.3482$)011.1(3000)011.1(3000

15

011.%1.1

3000

)1(

1515

0

0

P

t

r

P

rPP t

Exponential decay Exponential decay models are use to measure radioactive decay, decreasing populations, Half-life, and other elements that fit an exponential model. Again, the one variable in an exponential decay models in found in the exponent. Exponential Decay Formula

timet

rater

ValueOriginalP

ValueNewP

rPP t

0

0 )1(

Example 8 A certain population of black bears in the eastern United States has been decreasing by 3.1 percent per year. If this trend keeps up, what will be the population of bears in 20 years if there are currently 1000 bears.

533)969(.1000)031.1(1000

20

031.%1.3

1000

)1(

2020

0

0

P

t

r

P

rPP t

Example 9 A certain isotope decreases at a rate of 5% per year. It there is currently 340 grams of the isotope, how many grams of the isotope will there be in 20 years?

gramsP

t

r

P

rPP t

122)95(.340)05.1(340

25

05.%5

340

)1(

2020

0

0

Problem Set 1.6 Exponential Functions Evaluate using a calculator 1) 2e

2) e2

1

3) 3

4

2e Graph the following functions 4) xy 3

5) 1 xey

6) xey 2

7) xey 2 Growth Models (Show Work) 8) The current population of Germany is 80,000,000. What would be the population of Germany in 10 years if its population would growth at a steady rate of .9 % for 10 years? 9) The current population of Salem, Virginia is 25,000. What would be the population of Salem in 5 years if Salem would grow at a rate of 1.2 % per year? 10) Using the exponential growth formula, find the amount of money that you would have in a bank account if you deposited $10,000 in the account for 10 years at 1.6 % interest rate? 11) A certain rabbit population is modeled by the equation teP 03.2000 where t is the time in months. Use the model to predict the population after 20 months. Decay Models 11) A certain population of Panda Bears in China has been decreasing by 1.0 percent per year. If this trend keeps up, what will be the population of Panda Bears in 10 years if there are currently 2000 bears? 12) A certain isotope decreases at a rate of 4% per year. It there is currently 220 grams of the isotope, how many grams of the isotope will there be in 25 years?

Section 1.7 Basic Logarithms Definition of a Logarithm

abxa xb log

Example 1

i) Write 24335 as a logarithmic expression.

5243log2433 3

5

ii) Write 62554 as a logarithmic expression.

4625log6255 5

4

Example 2

i) Write 216log4 as exponential expression.

164216log 24

ii) Write 4000,10log10 as an exponential expression.

000,10104000,10log 4

10

Log base ten Another way of writing 1000log10 is 1000log .

The way we find the answer to 1000log is to ask the question of 10 raised to what power

gives you 1000? Since we know that 1000104 , the answer is 4.

Example 3

i) Find log 100,000 Since 000,100105 , 5000,100log ii) Find log 100 Since 100102 , 2100log

Example 4 Use a scientific calculator to evaluate the following logarithms

i) log 567 Answer: log 567 = 2.754 ii) log 30890 Answer: log 30890 = 4.490 iii) log 456782 Answer: log 456782 = 5.660

Graph of basic logarithms ________________________________________________________________________ Example 5 Graph xy 6log X Y 2 07.1)12log())2(6log( y 10 8.1)60log())10(6log( y 20 1.2)120log())20(6log( y 40 4.2)240log())40(6log( y

Plot the given values from the table gives the following graph

Example 6 Graph )1log(5 xy X y 2 4.2)3log(5)12log(5 y 10 2.5)11log(5)110log(5 y 20 6.6)21log(5)120log(5 y 40 1.8)41log(5)140log(5 y Plot the given values from the table gives the following graph

Example 7 (Using logarithmic models to model height) A logarithmic model to approximate the percentage P of an adult height a male has reached at an age A form 13 and 18 is 84)12log(16 AP

1) Sketch a graph of this function. P A 13 8484)1213log(16 P 14 8.88848.484)2log(1684)1214log(16 P 15 6.90846.784)3log(1682)1215log(16 P 18 5.96845.1284)6log(1684)1218log(16 P Plot the given values from the table gives the following graph

2) What does the graph tell you about the height of male after age of 18? Usually males stop growing after age 18 3) Use the model to compute the average height of a 16 year old male.

6.93846.984)4log(.1684)1216log(16 P 93.6% Example 8 Use the following model for $1000 invested in saving account given by

)log(4.2312.694 An , to find the amount of time (n) for the amount of money A to grow to $100,000.

8.462

11572.694

)5(4.2312.694

100000log4.2312.694

n

n

n

n

Problem Set 1.7

I) Write as a logarithmic expression. 1) ae 2 2) 6426

II) Write as an exponential expression. 1) 481log3

2) yx ln

III) Graph each logarithmic equation. 1) xy ln 2) 5ln xy

3) 62ln xy