Logarithmic and Exponential Functions

Logarithmic and Exponential Functions Studen

t Book - Series M-2

y

x

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Logarithmic and exponential functions Student Book - Series M 2 Contents Topics

Date completed

Topic 1 - The index laws

__ /__ /__

Topic 2 - Logarithms

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Topic 3 - Change of base

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Topic 4 - The functions y = ax and y = log(ax)

__ /__ /__

Topic 5 - The derivative of y = ax

__ /__ /__

Topic 6 - The number e and natural logarithms

__ /__ /__

Topic 7 - The derivative of y = ex

__ /__ /__

Topic 8 - The integral of ex

__ /__ /__

Topic 9 - The derivative of y = ln x

__ /__ /__

Topic 10 - The integral of 1x

__ /__ /__

Topic 11 - Applications of derivatives

__ /__ /__

Topic 12 - Applications of integrals of ex

__ /__ /__

Topic 13 - Applications of integration of 1x

__ /__ /__

Practice Tests Topic 1 - Topic test A

__ /__ /__

Topic 2 - Topic test B

__ /__ /__

Author of The Topics and Topic Tests: AS Kalra Logarithmic and exponential functions Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

ii

CHAPTER 4

Logarithmic andand exponential functionsfunctions Logarithmic exponential

EXCEL HSC MATHEMATICS page 128

Topic 1 - The index laws (1) The index laws (1) QUESTION 1

Use the index laws to simplify:

a

x2 × x5

b

a7 × a

c

4p12 ÷ 2p3

d

x2y3 × xy4

e

(a5)2

f

(3m2n)4

g

a0

h

6y0

i

x9 x3

j

6t8 2t 4

m (2a3b2)3 ÷ 4ab

QUESTION 2 a

23

e

42

1

QUESTION 3

k

n

x2 y5

l

x4 y4

15n9 ÷ 3n5 × 4n

o

2 ab 6 8 a 2b 6

(g4h3)2 × 2(gh2)3

Evaluate: b

104

f

83

2

c

31

d

60

g

25 2

h

320.8

c

4–4

d

10–5

g

16

h

1000

3

Write as fractions (in simplest form):

a

5–1

b

2–3

e

6–2

f

9

–1 2

–1 4

–2 3

95

CHAPTER 4 – Logarithmic and exponential functions

Logarithmic and exponential functions Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

1



Topic 1 - The index laws (2) The index laws (2) QUESTION 1 a

Use a calculator to find the value of:

87

e

2 455 489

QUESTION 2 2x

2.54

c

5291.5

f

5

g

(

371 293

3

0.027

)

4

d

2–7

h

3

(0.027) 4

Simplify:

a

5

÷ 5x

b

73x+4 × 79–3x

c

(3x)2 × (33x)3

d

82x ÷ 26x × 4x

e

93m+1 × 34m–1

f

32n ÷ 82n ÷ 43n

×5

3x

b

QUESTION 3

Solve:

a

k7 = 16 384

b

(25m)2 = 1 048 576

c

(1 – p)5 = 7776

d

93a = 37

e

5x × 253–x = 5

f

43q+5 × 82q–7 = 2

c

x2 × (2x3)3 = 15

QUESTION 4 a

8

x = 12 756

Find the value of x, correct to two decimal places, if: b

2x6 = 12.8

96

EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK


2



Topic 2 - Logarithms (1) Logarithms (1) QUESTION 1

Complete:

a

If loga x = c then x =

b

loga xy = loga x +

c

log a x y loga1 =

d

logaa =

f

loga xn =

e

QUESTION 2

Express as an integer:

a

log327

b

log232

c

log55

d

log71

e

log636

f

log10100 000

g

log7343

h

log2256

QUESTION 3

Simplify:

a

log62 + log63

b

log218 – log29

c

log 2 2

d

log520 + log52 – log58

e

log3504 – log37 – log38

f

log a 16 log a 4

c

4 logn3 – logn9

QUESTION 4 a

Express as a single logarithm:

3 loga2 + 2 loga3

b

logm12 + logm4 – logm8

97



3



Topic 2 - Logarithms (2) Logarithms (2) QUESTION 1

Complete: If y = ax then x =

QUESTION 2

Use a calculator to find the value, correct to three decimal places, of:

a

QUESTION 3 a

b

log1017

d

log100.35

b

c

loga10 000

logm10

b

logm8

c

logm2.5

d

logm12.5

b

log2x – log27 = 3

a

loga0.01

If logm2 = 0.289 and logm5 = 0.671, evaluate:

a

QUESTION 5

log101.65

If a2.37 = 10, find:

loga10

QUESTION 4

c

log10205

Find the value of x if:

log315 + log3x = log35

98



4



Change of base (1) Topic 3 - Change of base (1) QUESTION 1

Complete:

QUESTION 2

Simplify:

a

log927

d

2log816

QUESTION 3

log m a = log m b

b

log84

e

c

log432

log49 + log23

Find the value, correct to four decimal places, of:

a

log310

b

log715

c

log29

d

log411

e

log135

f

log689

g

log30.6

h

log20.75

i

log90.08

99



5



Topic 3 - Change of base (2) Change of base (2) QUESTION 1 a

Express as a logarithm to the given base:

log411 (base 2)

QUESTION 2

b

log256 (base 5)

c

log2732 (base 3)

Find the value of x, correct to three decimal places:

a

3x = 17

b

2x = 75

c

5x = 0.275

d

2(6x) = 45

e

7x–1 = 16

f

3 – 2x = 0.37

100



6


EXCEL HSC MATHEMATICS

Topic 4 - The functions y = a and y = log The functions y = a x and y = logaxa x

page 130

x

QUESTION 1 a

y=2

Sketch the graph of:

x

b

y = 7x

y

c y

x

QUESTION 2 a

y = 3–x y

x

x

Sketch the graph of: b

y = log10x y

c

y = log5x y

x

y = log2x y

x

x

QUESTION 3 a

On the same diagram sketch the graph of y = 3x and y = log3x

y

x

b

Complete: The graph of y = 3x and y = log3x are reflections of each other in the line 101



7



xx Topic 5 - The derivative of y = a

page 131

The derivative of y = a

Fill in the blanks in the derivation from first principles of y = 10x

QUESTION 1

f ( x + h )– f ( x ) dy = lim dx h → 0 h

= lim

h

h→0

= lim

x

10 (

)

h→0

= 10 x lim

h→0

QUESTION 2 a

)

Use a calculator to find, to two decimal places, the approximate value of:

 10 h – 1  lim  h  h→0 

QUESTION 3 a

(

h

b

 2h – 1  lim  h  h→0 

c

 3h – 1  lim  h  h→0 

c

ln 3

Find, to two decimal places: b

ln 10

ln 2

QUESTION 4

 ah – Use a calculator to find the value of a, to two decimal places, for which lim  h h→0 

QUESTION 5

 e h – 1 Using a calculator, find lim  h  h→0 

QUESTION 6

Write down the derivative of:

a

y = 5x

b

y = 7x

c

y = 4x

d

y = 11x

e

y = 6x

f

y = 9x

g

y = 8x

h

y = 15x

102

1  =1 



8

Logarithmic andand exponential functionsfunctions Logarithmic exponential Topic 6 - The number e and natural logarithms The number e and natural logarithms QUESTION 1

Write down the exact value of:

a

e0

b

ln 1

c

ln e

d

ln e2

e

eln2

f

7 ln e

g

ln e7

h

eln5

QUESTION 2

Find the value, correct to four decimal places, of:

a

e2

b

e4

c

2e5

d

e–1

e

ln 1.25

f

ln 7.8

g

loge3.6

h

ln 0.237

i

6 loge4

j

4e3 + 1

k

e

l

4 ln 3 – 1

QUESTION 3

Find the value of k, correct to three decimal places, if:

a

ek = 1.6

b

ln k = 1.9

c

3ek = 5.87

d

7e2k = 6

e

5e–4k = 3

f

10e3k+1 = 0.456

103



9



xx

page 131

Topic 7 - The derivative of y = e (1) The derivative of y = e (1) QUESTION 1

Find the derivative of:

a

y = ex

b

y = 3ex

c

f(x) = e2x

d

y = 4ex + 3

e

y = 2e5x

f

y = e–x

g

y = x – ex

h

y = 6e2x+5

i

f(x) = 4e–8x

j

y = 6x3 – 3e3x

k

y = ex – e–x

l

f(x) = 6 – 7e–9x

QUESTION 2

Use the product rule to differentiate:

a

y = xex

b

y = x2e2x

c

y = (3x – 4)e–x

d

y = 5e7x(x2 – 9x + 2)

104



10



x x (2) Topic 7 - The derivative of y = e

page 131

The derivative of y = e (2) QUESTION 1 a

y = (ex + 5)4

QUESTION 2 a

y=

b

f (x) = (4x – ex)3

b

the quotient rule

b

y=

Differentiate y = x , using: ex

the product rule

QUESTION 3 a


Differentiate:

ex x+1

3e x x2 – 5

105



11



x x Topic 8 - The integral of e

page 132

The integral of e QUESTION 1 a

∫e

d

∫e

g

∫

Find: b

∫ 5e

dx

e

∫

( e x + 2 x ) dx

h

∫

x

dx

2 x +3

QUESTION 2 a

d

g

∫

2

∫

–1

∫

2

0

–3

1

c

∫e

4 e – x dx

f

∫e

e 4 x dx 2

i

∫ (x

6 e x dx

c

1 e –2 x dx 2

f

x

dx

3x

dx

3–2 x

2

dx

– 8 x – 6 e –2 x ) dx

Find the exact value of:

e x dx

b

e 2 x+7 dx

e

( e x – e – x ) dx

h

∫

1

∫

2

∫

ln2

0

0

0

7 e x dx

106

i

∫

3

∫

3

∫

2

0

1

1

e 4 x dx

e 4– x dx

( e 3 x + x ) dx



12



Topic 9 - The derivative of y = ln x (1) The derivative of y = ln x (1) QUESTION 1

Differentiate:

a

y = loge x

b

y = loge 2x

c

f(x) = ln 6x

d

y = loge(7x + 5)

e

y = ln (1 – 2x)

f

y = ln (5x + 3)

g

y = ln x2

h

f(x) = ln x5

i

y = ln x9

j

f(x) = ln (x2 + 5)

k

y = ln (3x2 – 4)

l

y = loge (x3 – 7x2)

QUESTION 2 a


y = (ln x)2

b

f(x) = loge (3x – 1)2

107



13



Topic 9 - The derivative of y = ln x (2) The derivative of y = ln x (2) QUESTION 1 a

c

b

f(x) = ln x

QUESTION 2 a

Find the exact value of f′(e) if:

y = x ln x

ln x x2

c

f(x) = loge (2x – 1)

f(x) = 3 ln (x2 + 1)

Differentiate: b

d

y = x4 loge x

x+1 ln x

108



14

Logarithmic and exponential functions

Logarithmic and exponential functions 1


1 The integral of Topic 10 - The integral of x

x

QUESTION 1

Find:

a

∫

dx x

b

d

∫

2 x dx x +5

g

∫

8 x dx x2 – 3

2

QUESTION 2 a

c

∫

5

∫

3

2

0

∫

6 dx x

c

∫

e

∫

3 x 2 dx x3 – 2

f

∫ 3 x – 7 dx

h

∫

dx 4x – 1

i

∫ 1 – 2 x dx

3 dx x+2

3

7

Find the exact value of:

dx x–1

b

2 x dx x2 + 3

d

∫

e

∫

4

1

2

dx 2x

3x2 + 1 x3 + x

dx

109



15



Topic 11 - Applications of derivatives (1) Applications of derivatives (1) QUESTION 1

Find the equation of the tangent to the curve y = 2 ln x at the point where x = e

QUESTION 2

Find the equation of the normal to the curve y = 2e–x at the point where x = 1

QUESTION 3

The tangent to the curve y = ex at the point P meets the x-axis at an angle of 45°. Find the coordinates of P.

QUESTION 4

Find the maximum value of

ln x x

110



16



Topic 11 - Applications of derivatives (2) Applications of derivatives (2) QUESTION 1

Consider f(x) = ex(1 – x)

a

Where does the curve y = f(x) cross the x-axis?

b

Find any stationary points and determine their nature.

c

Find any points of inflexion.

d

Complete: i

e

as x → ∞, y →

ii as x → –∞, y →

Sketch the curve y = f(x) y

x

111



17



Topic 12 - Applications of integrals of exx

pages 135–136

Applications of integrals of e QUESTION 1 a

Find the area bounded by the curve y = ex, the x-axis, x = 0 and x = ln 3

y y = ex

3

b

0

Hence find the shaded area.

ln 3

x

QUESTION 2

A curve y = f(x) has a turning point at (0, 4). If f ″(x) = ex + e–x find the equation of the curve.

QUESTION 3

Show that the volume of the solid of revolution formed by rotating the curve y = ex, between x = 0 and x = 5 about the x-axis is given by π ( e 10 – 1) units3. 2 y y = ex

0

112

5

x



18



Logarithmic and exponential functions 1 Applications of integration of

pages 135–136

Topic 13 - Applications of integration of 1xx

QUESTION 1

Find the exact area bounded by the curve y = 4 , the x-axis and the ordinates x = 2 and x = 4 x y

x

QUESTION 2

The gradient function of a curve is given by 6 x – passes through the point (1, 7).

QUESTION 3

2 . Find the equation of the curve if it 2x – 1

Find the area shaded in the diagram.

y

y = 2 x

2 1 0

x

113



19


Logarithmic TOPIC TEST Topic Test

and exponential functions PART A

Time allowed: 1 hour Instructions

Total marks = 100

This part consists of 12 multiple-choice questions Each question is worth 1 mark

SECTION I Multiple-choice questions Calculators may be used

12 marks

Instructions This section consists of 12 multiple-choice questions Fill in only ONE CIRCLE for each question Each question is worth 1 mark Fill in only ONE CIRCLE Time allowed: 30 minutes Calculators may be used

Total marks = 12

1 23 × 2 2 = ? A 25

B

26

C

45

D

46

B

18

C

84

D

86

B

2

C

7

D

8

2 88 ÷ 8 2 = ? A 14

3 7m0 + 70 = ? A 1

4 p–3 = ? A

5

3

p

B

p3

C

1 p3

D

none of these

xn

C

xm xn

D

none of these

1

C

m2 m3

D

m2 3 m

m

xn =? A

6 m

–2 3

n

xm

B

m

=? 1

A

m3

B

3

m2

7 log42 = ? A

1 2

B

1

C

2

D

4

B

loga4.5

C

2 loga1.5

D

cannot be simplified

6.581

D

7.389

8 2 loga3 – loga2 = ? A loga7

9 The value of e2 correct to three decimal places is? A 0.301

B

0.693

C

114



20

Logarithmic and exponential functions Topic Test 10 10

PART A

d (e 2 x ) = ? dx d (e 2 x ) = ? 2x A dx e

C

2e2x

11 The could be aBsketch e2x 2ex of the graph of: C A diagram

2e2x

B

2ex

1 e2x 2 1 e2x 2

D y

x

y = 2 could be a sketch of the graph of: A diagram 11 The

D

y = f(x)

y

B y = 2x–x A

y = f(x)

1

–x x C y = 2 log B 2

2 ln2xx D y = log C D y = 2 ln x

01

1

2

x

0

1

2

x

12 log27 = ? ln 7 12 log A 27 = ? ln 2 ln 7 A ln 2

B B

ln 2 ln 7 ln 2 ln 7

C

2 ln 7

D

7 ln 2

C

2 ln 7

D

7 ln 2

Total marks achieved for PART A 12 88 marks

SECTION II

Topic Test SECTION II

PART 88 marksB

Show all necessary working

Show all necessary working 13 Simplify: Instructions Show all necessary working Time allowed: 30 minutes 2–x

b

log645 + log620 – log625

8x+1 × 25x ÷ 42–x

b

log645 + log620 – log625

x+1

a 8 ×2 13 Simplify: a

5x

÷4

14 Find x if: b

(1 – x)3 = 0.512

c

logx16 = 4

x8 = 1 679 616

b

(1 – x)3 = 0.512

c

logx16 = 4

15 Find, correct to three decimal places: a log b 15 Find, correct 102.9 to three decimal places: a

log102.9

1 mark each

1 mark each

a x8 = 1 679 616 14 Find x if: a

1 mark each Total marks = 88

b

1 mark each

1 mark each 5

9.3187

c

log211

5

9.3187

c

log211


Logarithmic and exponential functions CHAPTER 4 – Logarithmic and exponentialMathletics functions Instant Workbooks – Series M 2 Copyright © 3P Learning

1 mark each

115 11521

Logarithmic and exponential functions Topic Test

PART B

16 If loga3 = 0.565 and loga2 = 0.356 find:

1 mark each

a

b

loga6

c

loga9

loga1.5

17 Find the value of x, correct to three decimal places, if: a

5x = 424

b

2 marks each

1 – 3x = 0.57

c

6e2x+1 = 192

18 Write down the exact value of: a

1 mark each b

9 ln e

ln e4

c

eln8

19 Sketch the graph of: a

2 marks each

y = 8x

b

y = log8x

y

y

x

x


116

22

EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning


PART B

20 Differentiate: a

y=7

d

g

x

1 mark each b

y=e

y = 3e–2x

e

y = 5e7x–4

h

x

c

y = ln x

y = ln (5x – 4)

f

y = ln (x2 + 6x)

y = 4 loge(6 – 3x)

i

y = ex

21 Find the derivative of: a

y = x3e2x

c

e 6x – 1

6x

2

3 marks each b

y = 2x logex

d

ln x 4x + 1

Logarithmic and exponential functions CHAPTER 4 – Logarithmic and exponential functions Instant Workbooks – Series M 2 Copyright © 3P Learning Mathletics

117 23


PART B

22 Find:

2 marks each

a

∫

3 dx x

b

∫e

d

∫

e –2 x dx 2

e

∫ 2 x + 1 dx

8x

dx

3

c

∫ 2x

f

∫

23 Find the exact value of: a

c

e

∫

e

∫

5

∫

3

1

1

1

4 x dx 2 –3

1 e 5–3 x dx 4

3 marks each

1 dx x

b

dx ex

d

4 dx 7 – 2x

f

∫

1

∫

4

∫

2

0

1

0

e x dx 2

2 x + 5 dx x2 + 5x

e 3 x–4 dx


118

24


Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning


24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0

Topic Test

3 marks

at the point where x = 0

3 marks PART B

24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0

3 marks

25 Find the coordinates of the stationary point of the curve y = x ln x

4 marks


4 marks


4 marks

26 Find the area bounded by the curve y = 1 , the x-axis and the lines x = 1 and x = 5 x 26 Find the area bounded by the curve y = 1 , the x-axis and the lines x = 1 yand x = 5 x y 1 26 Find the area bounded by the curve y = , the x-axis and the lines x = 1 and x = 5 x

3 marks

24 Find the equation of the tangent to the curve y = 2e

x+1

3 marks 3 marks

y

x x –x

27 Find the volume of the solid of revolution formed when that portion of the curve y = e between x = –1 x and xthe = 1volume is rotated about 3 xmarks 27 Find of the solidthe of x-axis. revolution formed when that portion of the curve y = e–x between = –1 and x = 1 is rotated about the x-axis. 3 marks 27 Find the volume of the solid of revolution formed when that portion of the curve y y= e–x between x = –1 and x = 1 is rotated about the x-axis. 3 marks y = e–x y y = e–x

y

y = e–x


–1

0

1

x

–1

0

1

x

–1

0

1

x

Total marks achieved for PART B

119

88

119



CHAPTER 4 – Logarithmic and exponential functions Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning

119 25