Information
AI Chat

Chapter 10 Logarithms and polynomials

maths

Subject

Maths

108 Documents

Students shared 108 documents in this course

School

Penrith Selective High School

Academic year: 2021/2022

Uploaded by:

Anonymous Student

This document has been uploaded by a student, just like you, who decided to remain anonymous.

Penrith Selective High School

Recommended for you

Comments

Please sign in or register to post comments.

Preview text

####### Chapter

Logarithms

and

polynomials

10

####### What you will learn

10A Introducing logarithms 10B Logarithmic graphs 10C Laws of logarithms 10D Solving equations using logarithms 10E Polynomials 10F Expanding and simplifying polynomials 10G Dividing polynomials 10H Remainder theorem and factor theorem 10I Factorising polynomials to fi nd zeros 10J Graphs of polynomials

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

NSW Syllabus

####### for the Australian

####### Curriculum

S t r a n d : N u m b e r a n d A l g e b r a S u b s t r a n d s : L O G A R I T H M S P O LY N O M I A L S Outcomes A student recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems. (MA5–10NA) A student uses the defi nition of a logarithm to establish and apply the laws of logarithms. (MA5–11NA)

683

####### Modelling in fi nance

Economists and fi nancial analysts often deal with complex mathematical relationships that link variables such as cost, output, account balance and time. To model these relationships they use mathematical rules, including polynomials (e. linear, quadratic and cubic) and exponentials, the graphs of which give a visual representation of how the variables are related. Polynomials (i. sums of powers of x ) can be used when the x variable is in the base, as in the quadratic y = 2 x 2 − x + 1 or in the cubic P ( x ) = 3 x 3 − x 2 + 14. Exponentials combine with logarithms to model exponential growth and decay, and to solve complex problems involving compound interest, investment returns and insurance.

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

Number and Algebra 685

Introducing logarithms

Logarithms (‘logical arithmetic’) are an important idea in mathematics and were invented by John Napier in the 17th century to simplify arithmetic calculations. Logarithms are linked directly to exponentials and can be used to solve a range of exponential equations. Recall that 2 3 = 8 (2 to the power 3 equals 8). We can also say that the logarithm of 8 to the base 2 equals 3, which is written as log 28 = 3. So for exponential equations such as y = 2 x , a logarithm fi nds x for a given value of y. A logarithm can often be evaluated by hand, but calculators can also be used. Logarithms can also be used to create logarithmic scales, which are commonly used in science, economics and engineering. For example, the Richter scale, and the moment magnitude scale that replaced it, are logarithmic scales that measure the strength of an earthquake.

Let’s start: Can you work out logarithms? We know that 3 2 = 9, so log 39 = 2. This means that log 3 9 is equal to the index that makes 3 to the power of that index equal 9. Similarly 10 3 = 1000, so log 101000 = 3. Now fi nd the value of the following. - log 10100 • log 10 10 000 • log 216 - log 264 • log 327 • log 464

10A

Stage 5# 5. 5§ 5. 5◊ 5. 4

■ A logarithm of a number to a given base is the power (or index) to which the base is raised to give the number. - For example: log 216 = 4 since 2 4 = 16 - The base a is written as a subscript to the operator word ‘log’; i. log a... ■ In general, if ax = y then log ay = x with a > 0 and y > 0. - We say ‘log to the base a of y equals x ’.

####### Key ideas

Example 1Writing equivalent statements Write an equivalent statement to the following. a log 101000 = 3 b 25 = 32

S O L U T I O N E X P L A N AT I O N a 103 = 1000 log aayy = xx is equivalent to is equivalent to ax = y. b log 232 = 5 ax = y is equivalent to log aayy = x.

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

686 Chapter 10 Logarithms and polynomials

Example 2Evaluating logarithms a Evaluate the following logarithms. i log 28 ii log 5625 b Evaluate the following. i log 31 9

ii log 10 0. c Evaluate, correct to 3 decimal places, using a calculator. i log 107 ii log 10 0.

S O L U T I O N E X P L A N AT I O N a i log 28 = 3 Ask the question ‘2 to what power gives 8?’ Note: 2 3 = 8 ii log 5625 = 4 54 = 5 × 5 × 5 × 5 = 625

b i log 31 9

=− 2 3 − 2 =

1 3

1 29 =

ii log 10 0 =− 3 10 1 10

1 1000

− 3 == 33 = =0.

c i log 107 = 0 Use the log button on a calculator and use base 10. (Some calculators will give log base 10 by pressing the log button.)

ii log 10 0 =−0 Use the log button on a calculator.

Example 3Solving simple logarithmic equations Find the value of xx in these equations. in these equations. a log 464 = x b log 22 xx = 6

S O L U T I O N E X P L A N AT I O N a log 464 = x 4 x = 64 x = 3 (by inspection)

log aayy = xx then then ax = y.

43 = 64 b log 22 xx = 6 26 = x x = 64

Write in index form: 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

688 Chapter 10 Logarithms and polynomials

9 Find the value of x in these equations. a log 327 = x b log 232 = x c log 264 = x d log 5625 = x e log 101000 = x f log 636 = x g log 2 x = 4 h log 3 x = 4 i log 10 x = 3 j log 3 x = − 2 k log 4 x = − 1 l log 7 x = − 3 m log x 27 = 3 n log x 32 = 5 o log x 64 = 3 p log x 64 = 2 q log x 81 = 4 r log x 10 000 = 4 s log x 0 = − 1 t log 4 0 = x 10 A single bacterium cell divides into two every minute. a Complete this cell population table. b Write a rule for the population P after t minutes. c Use your rule to fi nd the population after 8 minutes. d Use trial and error to fi nd the time (correct to the nearest minute) for the population to rise to 10 000. e Write the exact answer to part d as a logarithm.

Example 3

Time (minutes) 0 1 2 3 4 5 Population 1 2

WORKING MA THE MATICAL

U F RCPS

WORKING MA THE MATICAL

U F 11 Is it possible for a logarithm (of the form log ab ) to give a negative result? If so, give an RCPS example and reasons. 12 If log ay = x and a > 0, is it possible for y ≤ 0 for any value of x? Give examples and reasons. 13 Evaluate: a log 24 × log 39 × log 416 × log 525

b 2 × log 327 − 5 × log 864 + 10 × log 101000

c 4 125 64

2 9 10

5 2

3 10

× + log × log

log log

14 Explain why 1 a 1log

does not exist.

Enrichment: Fractional logarithms

15 We know that we can write 2 2

1 = 2 , so log 2 2 1 2

= and log 232 1. 3

= Now evaluate the following without the use of a calculator. a log 242 b log 252 c log 3 3 d log 333 e log 7 7 f log 10310 g log 103100 h log 2316 i log 349 j log 5425 k log 2564 l log 3781

10A

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

Number and Algebra 689

Logarithmic graphs

The graph of the exponential equation y = ax is roughly a concave-up shape with the x -axis being an asymptote. If a mirror were placed along the line y = x , the refl ection of y = ax would be the graph of y = log a x , roughly a concave-down shape with the y -axis being an asymptote. Each point on the exponential graph, when refl ected, has its coordinates interchanged and the new point is on the logarithmic graph. Logarithm and exponential functions are the inverse of each other. Understanding logarithmic functions and their relationship to exponential functions enables mathematical techniques for solving a wide variety of problems, especially in engineering, science and computer technology. Log scales are used to measure physical quantities that increase exponentially. For example, the Richter scale is used to measure earthquake intensity; decibels measure the loudness of sound and pH measures the acidity of a solution.

Let’s start: Log scales Imagine that when you were born your grandparents invested $1500, compounding at a fi xed rate per annum for life. The value of your investment could be graphed either using a normal linear scale or a log scale, as shown by these graphs. Discuss the answers to the questions below.

- How do the coordinates of a point on the graph with the log scale relate to the corresponding point on the linear scale graph? Give an example. - What investment value does 3 correspond to on the log scale? - Which graph can be used to fi nd the investment value at age 20 years and what is this value? - By what factor has the investment value increased from age 20 to age 30? - By what factor has the investment value increased when the log scale increases by 1 unit? - If a graph using a log scale is linear, what form is the graph of the original data? - Discuss the advantages and disadvantages of linear and logarithmic scales.

10B

Stage 5# 5. 5§ 5. 5◊ 5. 4

0 10 20 30 40 Age (years)

Investment value vs age

50 60 70 80

$500 000

$1 000 000

$1 500 000

$2 000 000

$2 500 000

$3 000 000

$3 500 000

Log 10 (investment value) vs age

0 10 20 30 40 Age (years)

50 60 70 80

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

Number and Algebra 691

Example 4 Comparing the features of exponential and log graphs a Copy and complete these tables for y = 3 x and y = log 3 x.

x − 2 − 1 0 1 2 y = 3 x

x 1 9

1 3 1 3 9 y = log 3 x

b Plot the graphs of y = 3 x , y = log 3 x and also y = x on the same axes with − 3 ≤ x ≤ 3 and − 3 ≤ y ≤ 3. c State two features of y = log 3 x that show that it is the inverse function of y = 3 x. d For each graph, state the equation of its asymptote. e State any limitations for the input values for y = 3 x and y = log 3 x. f State any limitations for the output values for y = 3 x and y = log 3 x.

S O L U T I O N E X P L A N AT I O N a x − 2 − 1 0 1 2

y = 3 x 1 9

1 3 1 3 9

x 1 9

1 3 1 3 9

y = log 3 x − 2 − 1 0 1 2

3 1 3

2 2

− = = 1 9

log 3 2 2. 1 9 2 3 1 3

1 9

 

  = − since − = =

b y

y = log 3 x

y = x y = 3 x

x

–3–2 3

–1 12

Plot points and join them with a smooth curve.

c i y = log 3 x is the mirror image of y = 3 x , refl ected in the line y = x. ii When the coordinates of a point on y = 3 x are reversed, it gives the coordinates of a point on y = log 3 x.

Inverse functions are refl ected in the line y = x. By drawing lines perpendicular to y = x , the mirror image from each point on y = 3 x can be found and it will have coordinates in the reverse order; e. (1, 3) is on y = 3 x and (3, 1) is its mirror image on y = log 3 x.

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

692 Chapter 10 Logarithms and polynomials

d y = 3 x has asymptote y = 0. y = log 3 x has asymptote x = 0.

As x → −∞, 3 x → 0; i. the curve approaches the x -axis ( y = 0) but never reaches it. As x → 0, log 3 x → −∞; i. the curve approaches the y -axis ( x = 0) but never reaches it. e y = 3 x has no limitations on x values_. x_ > 0 for y = log 3 x.

Any x value input into y = 3 x produces a valid output. Only positive x values input into y = log 3 x give a valid output.

f y > 0 for y = 3 x. There are no limitations on the y values for y = log 3 x.

The output values ( y ) from y = 3 x are always positive. The output values ( y ) from y = log 3 x can result in any real number value.

Example 5 Identifying transformations, asymptotes and intercepts of log graphs The log equation y = log 2 ( x − 3) + 1 is graphed here. a State any limitations on the input values (i. x values). b State the equation of the asymptote. c State the horizontal and the vertical translations that have been applied to y = log 2 x. d Determine the axes intercepts, if any.

S O L U T I O N E X P L A N AT I O N a x > 3 ( x − 3) > 0, x > 3 If x ≤ 3 the log calculation will be invalid as there is no result for the log of a negative number. b x = 3 Asymptote equation when ( x − 3) = 0, so x = 3. An asymptote is a line that the curve approaches by getting closer and closer to it, but never reaching it. c Horizontal translation of 3 units to the right. Vertical translation of 1 unit up.

The asymptote equation x = 3 shows that the curve has moved 3 units to the right, increasing each x value by 3 units. Adding 1 to the log function means the curve moves up by 1 unit so each y value is increased by 1 unit.

4 3

O 1 4 5 6 7 8 92 103

y = log 2 ( x – 3) + 1

x = 3

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

694 Chapter 10 Logarithms and polynomials

b y

x

2 1

(–1, –1) –

O 1 2 3 4 5–1–

x = –

(0, –0)

Asymptote is when ( x + 2) = 0 so x = −2, showing the curve moves 2 units to the left; h = −2 and k = −1. A point on the graph is (1 + h , k ) = (−1, −1) log ( ) log ( )

5 5 1

2 1 0 2 1 2 5 3

x x x x

− =
- =
  - =x -intercept is 3. y y

= + − = −

= −

log ( ) log log log

5 5 10 10

0 2 1 2 1 2 5

1 See nnote below.)

y = −0 5693. ... y -intercept is −0 (to 2 decimal places). #In this step we used the change of base formula, which is explained in Section 10C. Rule: log log a log x x a

= 10 10 Example: log log 5 log 10 10

2 2 5

1 For the graph of y = log 2 x , state: a any limitations on the input values (i. x values) b any limitations on the output values (i. y values) c how y changes as x approaches zero d the equation of the asymptote e the x -intercept 2 State two features of y = log 2 x which show that it is the inverse function of y = 2 x. 3 For the graph of y = log 2 ( x − 4) + 5, state: a any limitations on the value of x b the equation of the asymptote c the horizontal translation of y = log 2 x d the vertical translation of y = log 2 x

–1–

O 1 3 4 52

y = log 2 x

Exercise 10B WORKING MA THE MATICAL

U F RCPS

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

Number and Algebra 695

4 a Copy and complete this table, showing the log scale values that would be used for the given physical values.

Physical values 1 10 100 1000 10 000 100 000 log scale log 10 (physical value)

b When the physical value increases by a factor of 10, what is the increase in the log scale? c By what factor has the physical value increased when the log scale increases by 2 units? d What log scale value would be used for a physical value of 16 000? e What physical value would be represented by 0 on the log scale? f What physical value would be represented by 0 on the log scale? g What log scale value would be used for a physical value of 2 3? h If the physical value doubles, by how much does the log scale increase? i What log scale value would be used for a physical value of 2 10?

WORKING MA THE MATICAL

U F RCPS 5 a Copy and complete these tables for y = 2 x and y = log 2 x.

x − 3 − 2 − 1 0 1 2 3 y = 2 x

x 81 1 4

1 2 1 2 4 y = log 2 x b Plot the graphs of y = 2 x , y = log 2 x and also y = x on the same axes with − 4 ≤ x ≤ 4 and − 4 ≤ y ≤ 4. c State two features of y = log 2 x which show that it is the inverse function of y = 2 x. d For each graph, state the axes intercepts, if any. e For each graph, state the equation of its asymptote. f State any limitations for the input values for y = 2 x and y = log 2 x. g State any limitations for the output values for y = 2 x and y = log 2 x. 6 a Copy and complete these tables for y = 10 x and y = log 10 x.

x − 3 − 2 − 1 0 1 2 3 y = 10 x

x 0 0 1 10 100 y = log 10 x b Graph y = 10 x , y = log 10 x and also y = x on the same axes with − 2 ≤ x ≤ 10 and − 2 ≤ y ≤ 10. c For each graph, state the axes intercepts, if any.

Example 4

WORKING MA THE MATICAL

U F RCPS

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

Number and Algebra 697

a Calculate the Richter scale magnitude of an earthquake with intensity I = 10 000 I 0. b How much more intense than I 0 is an earthquake measuring 7 on the Richter scale? c Rearrange the equation to make I the subject. d Show mathematically that an earthquake of magnitude 7 is 1000 times more intense than an earthquake of magnitude 4. e Show mathematically that when one earthquake is double the intensity of another, their difference in magnitude on the Richter scale is 0. f How much stronger in intensity was the 2011 East Japan earthquake, of magnitude 9, compared with the 2011 Christchurch, New Zealand earthquake, of magnitude 6? g The most damaging Australian earthquake was in 1989, Newcastle, NSW, measuring 5. It killed 13 people, injured 160 and did $4 billion worth of damage. The strongest onshore Australian earthquake was in 1941, Meeberrie, WA, measuring 7 but caused only minor damage. How much more intense was the energy of the Meeberrie earthquake and why did it cause so much less damage than the Newcastle earthquake? 11 A logarithmic scale is used to measure the loudness of sound with units in decibels.

dB=

 



 

10 10  0

log I I

, where dB is in decibels, I is the intensity

of the sound and I 0 is the threshold of sound intensity detected by the human ear. a What is the decibel rating of sound at the threshold of hearing (i. I = I 0 )? b Find the decibel rating of normal conversation, which is 1 million times louder than the threshold of human hearing. c Calculate how many times louder a hair dryer is at 80 dB compared with the threshold of human hearing. d How much greater is the noise intensity of an iPod playing at a peak volume of 115 dB compared with a half volume level of 85 dB? e By what factor does the noise intensity increase for each 3 dB increase? (Round your answer to the nearest whole number.) f The human ear can withstand 85 dB for 8 hours before permanent hearing damage occurs. For every 3 dB increase over 85 dB the safe exposure time is halved. Calculate the safe exposure time to these noises: Band practice 91 dB; belt sander 94 dB; hand drill 97 dB; chainsaw 100 dB; rock concert 112 dB; iPod earphones at peak volume 115 dB; jet engine at take-off 140 dB.

WORKING MA THE MATICAL

U F RCPS

ISBN 978-1-107-67670- © David Greenwood et al. 2014 Cambridge University Press

698 Chapter 10 Logarithms and polynomials

12 The pH of a solution is a measure of its acidity or alkalinity. Pure water has a pH of 7 and is neutral, whereas acids have pH < 7 and alkaline (basic) solutions have pH > 7. pH = −log 10 [H+], where [H+] is the concentration of hydrogen ions, in moles/litre. a Calculate the pH of the following liquids and state whether they are acidic or alkaline. Round to 1 decimal place where necessary. i lemon juice [H+] = 3 × 10 − 3 moles/litre ii beer [H+] = 1 × 10 − 4 moles/litre iii blood [H+] = 4 × 10 − 8 moles/litre iv hand soap [H+] = 1 × 10 − 10 moles/litre b Rearrange the pH equation, making [H+] the subject. c By what factor has the hydrogen ion concentration increased in black coffee (pH = 5) compared with pure water (pH = 7)? d By what factor does the hydrogen ion concentration change when the pH increases by 1? e The pH of a cola drink is 2. If a soft drink has half the acidity of cola, what is its [H+] and pH? f If the ocean absorbs enough CO 2 to cause the pH of sea water to change from 8 to 7, what percentage increase in acidity (i. [H+]) would this cause? Round your answer to the nearest whole number.

WORKING MA THE MATICAL

U F RCPS

13 a Using technology, graph each pair of functions on the same axes. i y = 2 x , y = log 2 x ii y = 5 x , y = log 5 x iii y = 12 x , y = log 12 x b Compare logarithmic graphs to exponential graphs; that is, state any similarities between the two graph types. c Contrast logarithmic graphs to exponential graphs; that is, state any differences between the two graph types. 14 Find the rule for each logarithmic graph in the form y = log a ( x − h ) + k. a

–1–2–3–4 –1 1 2 3 4 5 6 - -

2 (–1, 1)

(5, 3) 3

x O

–1–1 1 2 3 4

(3, 3)

(6, 5)

5 6

O 7 8 9 10

WORKING MA THE MATICAL

U F RCPS

10B

700 Chapter 10 Logarithms and polynomials

Laws of logarithms

From the study of indices you will recall a number of index laws that can be used to manipulate expressions involving powers. Similarly, we have laws for logarithms and these can be derived using the index laws. Recall this index law: am × an = am + n Now let x = am and y = an (1) so m = log ax and n = log ay (2) From equation (1) xy = am × an = am + n So: m + n = log a ( xy ) From (2) m + n = log ax + log ay So: log a ( xy ) = log ax + log ay This is a proof for one of the logarithm laws. We will develop the others later in this section.

Let’s start: Proving a logarithm law In the introduction above there is a proof of the fi rst logarithm law, which is considered in this section. It uses the index law for multiplication. - Now complete a similar proof for the second logarithm law, log a log a log a , x y x y  

  = − using the index law for division.

10C

Stage 5# 5. 5§ 5. 5◊ 5. 4

■ log a ( xy ) = log ax + log ay - This relates to the index law am × an = am + n.

■ log a x log a log a y

 x y 

 

= −

- This relates to the index law am ÷ an = am − n. ■ log a ( xn ) = n log ax - This relates to the index law ( am ) n = am × n. ■ Other properties of logarithms: - log a 1 = 0, using a 0 = 1 - log aa = 1, using a 1 = a - log a 1 x

= log ax − 1 = −log ax.

####### Key ideas

Example 7 Simplifying logarithmic expressions Simplify the following. a log a 4 + log a 5 b log a 22 − log a 11 c 3log a 2

Number and Algebra 701

S O L U T I O N E X P L A N AT I O N a log a 4 + log a 5 = log a 20 This uses the law log a ( xy ) = log ax + log ay.

b log a 22 − log a 11 = log a 2 This uses the law log a x log a log a y

 x y 

 

= −. Note: log a 22 log a 11

= 2

c 3log a 2 = log a 23 = log a 8

This uses the law n log ax = log axn.

Example 8 Evaluating logarithmic expressions Simplify and evaluate the following expressions. a log 21 b log 55 c log 61 36

d log 26 − log 23

S O L U T I O N E X P L A N AT I O N a log 21 = 0 20 = 1 b log 55 = 1 51 = 5

c log 61 log 6 2 36

= 6 − = − 2 × log 66 = − 2 × 1 = − 2

Alternatively, use the rule log a x

1 = −log ax. So log 6 log 6 1 36

36 2

=− = −

d log 26 − log 23 = log 22 = 1

log 2 6 log 2 3

 = 2

and 2 1 = 2

1 Copy and complete the rules for logarithms using the given pronumerals. a log b ( xy ) = log bx + _________

b log b x y

 

 

= _______ − _______

c log abm = m × _______ d log aa = _______ e log c 1 = _______ f log a b

1 = _______

Exercise 10C WORKING MA THE MATICA LLY

U F R PS C

Was this document helpful?