Logarithmic and Exponential...

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Logarithmic and Exponential Functions

Student Book - Series M-2

y

x

Logarithmic and exponential functionsStudent Book - Series M 2

Contents

Topics

Practice Tests

Topic 1 - The index laws __ /__ /__

Topic 1 - Topic test A __ /__ /__

Topic 2 - Logarithms __ /__ /__

Topic 10 - The integral of 1 x __ /__ /__

Topic 2 - Topic test B __ /__ /__

Topic 3 - Change of base __ /__ /__

Topic 11 - Applications of derivatives __ /__ /__

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

Topic 12 - Applications of integrals of ex __ /__ /__

Topic 5 - The derivative of y = ax __ /__ /__

Topic 13 - Applications of integration of 1 x __ /__ /__

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 __ /__ /__

Date completed

Author of The Topics and Topic Tests: AS Kalra

iiLogarithmic and exponential functions

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

1

Logarithmic and exponential functions

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

Topic 1 - The index laws (1)

95CHAPTER 4 – Logarithmic and exponential functions

CHAPTER 4

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

xx

9

3j

6

2

8

4

t

tk

x yx y

2 5

4 4l

2

8

6

2 6

ab

a b

m (2a3b2)3 ÷ 4ab n 15n9 ÷ 3n5 ! 4n o (g4h3)2 ! 2(gh2)3

QUESTION 2 Evaluate:

a 23 b 104 c 31 d 60

e 412 f 8

23 g 2

325 h 320.8

QUESTION 3 Write as fractions (in simplest form):

a 5–1 b 2–3 c 4–4 d 10–5

e 6–2 f 9– 1

2 g 16– 1

4 h 1000–23

Logarithmic and exponential functionsEXCEL HSC MATHEMATICS

page 128

2



Topic 1 - The index laws (2)

96 EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK

The index laws (2)

QUESTION 1 Use a calculator to find the value of:

a 87 b 2.54 c 5291.5 d 2–7

e 2 455 489 f 371 2935 g

0.02734( ) h (0.027)43

QUESTION 2 Simplify:

a 52x ! 53x ÷ 5x b 73x+4 ! 79–3x c (3x)2 ! (33x)3

d 82x ÷ 26x ! 4x e 93m+1 ! 34m–1 f 32n ÷ 82n ÷ 43n

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

QUESTION 4 Find the value of x, correct to two decimal places, if:

a x8 = 12 756 b 2x6 = 12.8 c x2 ! (2x3)3 = 15


page 128

3



Topic 2 - Logarithms (1)


Logarithms (1)

QUESTION 1 Complete:

a If loga x = c then x = b loga xy = loga x +

c loga

xy

d logaa =

e loga1 = f loga xn =

QUESTION 2 Express as an integer:

a log327 b log232 c log55 d log71

e log636 f log10100 000 g log7343 h log2256


a log62 + log63 b log218 – log29 c log 22

d log520 + log52 – log58 e log3504 – log37 – log38 f

log 16log 4

a

a

QUESTION 4 Express as a single logarithm:

a 3 loga2 + 2 loga3 b logm12 + logm4 – logm8 c 4 logn3 – logn9


page 129

4



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 log1017 b log10205 c log100.35 d log101.65

QUESTION 3 If a2.37 = 10, find:

a loga10 b loga10 000 c loga0.01

QUESTION 4 If logm2 = 0.289 and logm5 = 0.671, evaluate:

a logm10 b logm8

c logm2.5 d logm12.5

QUESTION 5 Find the value of x if:

a log315 + log3x = log35 b log2x – log27 = 3


page 129

5



Topic 3 - Change of base (1)


Change of base (1)

QUESTION 1 Complete:

loglog

=m

m

ab


a log927 b log84 c log432

d 2log816 e log49 + log23

QUESTION 3 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


page 130

6



Topic 3 - Change of base (2)


Change of base (2)

QUESTION 1 Express as a logarithm to the given base:

a log411 (base 2) b log256 (base 5) c log2732 (base 3)

QUESTION 2 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


page 130

7



Topic 4 - The functions y = ax and y = logax


The functions y = a x and y = logax

QUESTION 1 Sketch the graph of:

a y = 2x b y = 7x c y = 3–x

QUESTION 2 Sketch the graph of:

a y = log10x b y = log5x c y = log2x

QUESTION 3

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

b Complete: The graph of y = 3x and y = log3x are reflections of eachother in the line

y

x

y

x

y

x

y

x

y

x

y

x

y

x


page 130

8



Topic 5 - The derivative of y = ax


The derivative of y = ax

QUESTION 1 Fill in the blanks in the derivation from first principles of y = 10x

dydx

f x h f xhh

= lim ( + )– ( )

0!

= lim

0h h!

= lim

10 ( )0h

x

h!

= 10 lim

0

x

h! ( )QUESTION 2 Use a calculator to find, to two decimal places, the approximate value of:

a lim

10 – 10h

h

h!

"

#$%

&'b

lim

2 – 10h

h

h!

"

#$%

&'c

lim

3 – 10h

h

h!

"

#$%

&'

QUESTION 3 Find, to two decimal places:

a ln 10 b ln 2 c ln 3

QUESTION 4 Use a calculator to find the value of a, to two decimal places, for which lim

– 1 = 1

0h

hah!

"

#$%

&'

QUESTION 5 Using a calculator, find lim

– 10h

heh!

"

#$%

&'

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


page 131

9



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


10



Topic 7 - The derivative of y = ex (1)


The derivative of y = ex (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)


page 131

11



Topic 7 - The derivative of y = ex (2)


The derivative of y = ex (2)


a y = (ex + 5)4 b f (x) = (4x – ex)3

QUESTION 2 Differentiate y x

e x = , using:

a the product rule b the quotient rule

QUESTION 3 Differentiate:

a y e

x

x =

+ 1b

y

e

x

x =

3

– 52


page 131

12



Topic 8 - The integral of ex


The integral of ex

QUESTION 1 Find:

a ! e dxx b

! 5e dxx c ! e dxx3

d ! e dxx2 +3 e

! 4 –e dxx f ! e dxx3–2

g ! ( + 2 )e x dxx h

!e dx

x4

2 i

! ( – 8 – 6 )2 –2x x e dxx

QUESTION 2 Find the exact value of:

a 0

2

! e dxx b 0

16! e dxx c

0

34! e dxx

d –3

–12 +7! e dxx e

0

2–21

2! e dxx f 1

34–! e dxx

g 1

2–( – )! e e dxx x h

0

ln27! e dxx i

1

23( + )! e x dxx


page 132

13



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


The derivative of y = ln x (1)


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)


a y = (ln x)2 b f(x) = loge (3x – 1)2


page 133

14



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


The derivative of y = ln x (2)

QUESTION 1 Find the exact value of f !(e) if:

a f(x) = ln x b f(x) = loge (2x – 1) c f(x) = 3 ln (x2 + 1)


a y = x ln x b y = x4 loge x

c

ln2

x

xd

x

x + 1ln


page 133

15




The integral of 1x

QUESTION 1 Find:

a !

dxx

b !

6 x

dx c !

3 + 2

x

dx

d ! 2

+ 5

2x

xdx e

! 3

– 2

2

3x

xdx f

!3

3 – 7

xdx

g ! 8

– 3

2x

xdx h

!dxx4 – 1

i !

71 – 2

x

dx

QUESTION 2 Find the exact value of:

a 2

5

– 1! dxx

b 1 2

e dxx!

c 0

3

22 + 3

! xx

dx d 2

4 2

3

3 + 1

+ ! x

x xdx


page 134

Topic 10 - The integral of 1 x

16




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 thecoordinates of P.

QUESTION 4 Find the maximum value of

ln xx


page 135Topic 11 - Applications of derivatives (1)

17




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 as x ! ", y ! ii as x ! –", y !

e Sketch the curve y = f(x)


page 135

x

y

Topic 11 - Applications of derivatives (2)

18




Applications of integrals of ex

QUESTION 1

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

b Hence find the shaded area.

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 "2

( – 1)10e units3.


pages 135–136

x50

y = ex

y

y = ex

xln 30

3

y

Topic 12 - Applications of integrals of ex

19




Applications of integration of 1x

QUESTION 1 Find the exact area bounded by the curve y

x = 4 , the x-axis and the ordinates x = 2 and x = 4

QUESTION 2 The gradient function of a curve is given by 6 – 2

2 – 1x

x. Find the equation of the curve if it

passes through the point (1, 7).

QUESTION 3 Find the area shaded in the diagram.

y

x

x0

1

2

y

y

x = 2


pages 135–136

Topic 13 - Applications of integration of 1 x


TOPIC TEST

Time allowed: 1 hour Total marks = 100

SECTION I Multiple-choice questions 12 marksInstructions This section consists of 12 multiple-choice questions

Each question is worth 1 markFill in only ONE CIRCLECalculators may be used

1 23 ! 22 = ?

A 25 B 26 C 45 D 46

2 88 ÷ 82 = ?

A 14 B 18 C 84 D 86

3 7m0 + 70 = ?

A 1 B 2 C 7 D 8

4 p–3 = ?

A p3 B p3 C

13p

D none of these

5 xmn = ?

A x mn B x nm C

xx

m

nD none of these

6 m– 2

3 = ?

A

13m

B

123 m

C

mm

2

3D

mm

2

3

7 log42 = ?

A 12

B 1 C 2 D 4

8 2 loga3 – loga2 = ?

A loga7 B loga4.5 C 2 loga1.5 D cannot be simplified

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

A 0.301 B 0.693 C 6.581 D 7.389


20Logarithmic and exponential functions


Instructions This part consists of 12 multiple-choice questions Each question is worth 1 mark Calculators may be used Fill in only ONE CIRCLE for each question

Time allowed: 30 minutes Total marks = 12

Logarithmic and exponential functionsTopic Test PART A


10 ddx

e x ( ) = ?2

A e2x B 2ex C 2e2x D 12 2e x

11 The diagram could be a sketch of the graph of:

A y = 2x

B y = 2–x

C y = log2x

D y = 2 ln x

12 log27 = ?

A ln 7ln 2

B ln 2ln 7

C 2 ln 7 D 7 ln 2

SECTION II 88 marksShow all necessary working

13 Simplify: 1 mark each

a 8x+1 ! 25x ÷ 42–x b log645 + log620 – log625

14 Find x if: 1 mark each

a x8 = 1 679 616 b (1 – x)3 = 0.512 c logx16 = 4

15 Find, correct to three decimal places: 1 mark each

a log102.9 b 9.31875 c log211

0 1 2 x

y = f(x)

1

y


10 ddx

e x ( ) = ?2

A e2x B 2ex C 2e2x D 12 2e x

11 The diagram could be a sketch of the graph of:

A y = 2x

B y = 2–x

C y = log2x

D y = 2 ln x

12 log27 = ?

A ln 7ln 2

B ln 2ln 7

C 2 ln 7 D 7 ln 2

SECTION II 88 marksShow all necessary working

13 Simplify: 1 mark each

a 8x+1 ! 25x ÷ 42–x b log645 + log620 – log625

14 Find x if: 1 mark each

a x8 = 1 679 616 b (1 – x)3 = 0.512 c logx16 = 4

15 Find, correct to three decimal places: 1 mark each

a log102.9 b 9.31875 c log211

0 1 2 x

y = f(x)

1

y

Topic Test PART BInstructions Show all necessary working Time allowed: 30 minutes Total marks = 88

12



Logarithmic and exponential functionsTopic Test PART A

Total marks achieved for PART A



Logarithmic and exponential functionsTopic Test PART B


16 If loga3 = 0.565 and loga2 = 0.356 find: 1 mark each

a loga6 b loga9 c loga1.5

17 Find the value of x, correct to three decimal places, if: 2 marks each

a 5x = 424 b 1 – 3x = 0.57 c 6e2x+1 = 192

18 Write down the exact value of: 1 mark each

a 9 ln e b ln e4 c eln8

19 Sketch the graph of: 2 marks each

a y = 8x b y = log8x

x

y

x

y



Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning 117CHAPTER 4 – Logarithmic and exponential functions

20 Differentiate: 1 mark each

a y = 7x b y = ex c y = ln x

d y = 3e–2x e y = ln (5x – 4) f y = ln (x2 + 6x)

g y = 5e7x–4 h y = 4 loge(6 – 3x) i y e x = 2

21 Find the derivative of: 3 marks each

a y = x3e2x b y = 2x logex

c

ex

x6

6 – 1d

ln

4 + 1x

x



Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning118 EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK

22 Find: 2 marks each

a ! 3

xdx b

! e dxx8 c ! 4

2 – 32x

xdx

d !

e dxx–2

2 e

!3

2 + 1

xdx f

!14 5–3e dxx

23 Find the exact value of: 3 marks each

a 1

1 e

xdx! b

0

1

2 ! e dx

x

c 1

5

! dxe x

d 1

4

22 + 5

+ 5 ! x

x xdx

e 1

3 47 – 2

! xdx f

0

23 –4! e dxx

Total marks achieved for PART B




88


x

y

y = e–x

y

0 1–1 x

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

26 Find the area bounded by the curve y

x = 1 , the x-axis and the lines x = 1 and x = 5 3 marks

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


x

y

y = e–x

y

0 1–1 x







x

y

y = e–x

y

0 1–1 x






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