Exponential and Logarithmic Functions · PDF file · 2014-10-20Solve the following simultaneous equations: ... Find the values of the following common logarithms without using a calculator

1 1

Exponential and Logarithmic Functions

指數及對數函數

Exercises(練習)

1. Express each of the following in the form px , where p is a rational number.

把下列各題寫成 px 的形式，其中 p 是一個有理數。

(a) 7 4x

(b) 35 )(

1

x

(a)

7

4

7

1

47 4 )(

x

xx

(b)

5

3

5

3

35

135

1

)(

1

)(

1

x

x

xx

2. Simplify 42

23

)2(

)6(

x

x and express your answer with positive indices.

化簡 42

23

)2(

)6(

x

x，並以正指數表示答案。

2

2

86

8

6

424

232

42

23

4

9

4

9

4

9

16

36

)(2

)(6

)2(

)6(

x

x

x

x

x

x

x

x

x

Question Bank

2

3. Find the values of

求下列各題的值。

(a) 4 16 ,

(b) 3 27 .

(a) ∵

162

162222

4

∴ 2164

(b) ∵ 27)3()3()3(

27)3( 3

∴ 3273

4. Simplify

2

32

53

yx

yxand express your answer with positive indices.

化簡

2

32

53

yx

yx，並以正指數表示答案。

235)2(3

2

32

53

][

yx

yx

yx

10

16

1610

285 )(

x

y

yx

yx



(a) 3

1

)125( ,

(b) 2

1

81

.

(a)

5

])5[()125( 3

1

33

1

5 Exponential and Logarithmic Functions

3

(b)

9

1

)9(

])9[(

)81(81

1

12

1

2

12

1

2

1



(a) 5

3

)32(

,

(b) 3

2

125

64

.

(a)

8

1

)2(

1

)2(

)2(

])2[()32(

3

3

5

35

5

3

55

3

(b)

16

25

25

16

5

4

5

4

125

64

125

64

1

12

1

3

2

3

1

3

2

3

2

Question Bank

4

7. Solve 3055 1 xx .

解 3055 1 xx 。

2

1

1

1

55

255

30)5(5

6

30)15

1(5

30)15(5

305)5(5

3055

x

x

x

x

x

xx

xx

∴ 2x

8. Simplify the following expressions and express your answers with positive indices.

化簡下列各題，並以正指數表示答案。

(a) x

xx

24 3

(b) 2

1

2

1

4 3

3 2

)9(

xx

x

(a)

2

1

2

4

1

32

4 3 )(

x

xx

x

xx

2

12

4

3

xx

2

5

4

3

xx

4

7

4

7

4

103

2

5

4

3

1

x

x

x

x


5

(b)

6

1

)4

1(

4

3

3

2

4

1

1

4

3

3

2

4

1

12

1

2

4

3

3

2

4

1

12

1

4

3

3

2

)2

1(

2

1

2

1

4

1

3

3

1

22

1

2

1

4 3

3 2

3

3

3

])3[(

)9(

9

)(

)()9(

x

x

x

x

x

x

x

x

x

x

x

x

x

xx

x

x

9. Solve 3

5

816 x .

解 3

5

816 x 。

54

22

)2()2(

816

54

3

5

34

3

5

x

x

x

x

∴ 4

5x

10. Solve the following simultaneous equations:

解以下的聯立方程：

1282

81332

23

yx

yx

)2......(1282

)1......(81332

23

yx

yx

From (1),

423 33 yx

∴ 423 yx

From (2),

732 22 yx

∴ 732 yx

Consider the simultaneous equations:

Question Bank

6

)4......(732

)3......(423

yx

yx

3 (3) + 2 (4),

2

2613

1412)64()69(

x

x

yxyx

By substituting x = 2 into (3), we have

1

22

426

42)2(3

y

y

y

y

∴ The solution is x = 2, y = 1.

11. Solve 01)2(32 12 xx .

解 01)2(32 12 xx 。

01)2(3)2(2

01)2(32

2

12

xx

xx

Let xy 2 , the equation becomes

0)1)(12(

0132 2

yy

yy

2

1y or 1y

∴ 2

12 x or 12 x

122 x or 022 x

∴ 1x or 0x

12. In each of the following, find the value of x correct to 3 significant figures.

求下列各題中 x 的值，準確至三位有效數字。

(a) 410 x

(b) 710 x

(a) ∵ 410 x

∴

602.0

4log

x (cor. to 3 sig. fig.)

(b) ∵ 710 x

∴

845.0

7log

x

(cor. to 3 sig. fig.)


7

13. Find the values of the following common logarithms without using a calculator.

試不使用計算機，求下列各常用對數的值。

(a) log 10 000 000

(b) log 0.000 1

(a) ∵ 10 000 000 = 107

∴ log 10 000 000 = 7

(b) ∵ 0.000 1 = 10–4

∴ log 0.000 1 = 4

14. Find the values of the following logarithms without using a calculator.

試不使用計算機，求下列各對數的值。

(a) 32log2

(b) 243log3

(a) ∵ 3225

∴

532log2

(b) ∵ 24335

∴

5243log3

15. Find the values of the following expressions without using a calculator.


(a) 3 7log

49log

(b) 216log

36log

7

7

(a)

6

3

1

2

7log3

1

7log2

7log

7log

7log

49log

3

1

2

3

Question Bank

8

(b)

3

4

2

3

2

6log2

3

6log2

6log

6log

)6(log

6log

216log

36log

7

7

2

3

7

27

2

1

37

27

7

7

16. Find the values of the following expressions without using a calculator.


(a) 40log25log

(b) 7log189log 33

(a)

3

10log3

10log

1000log

)4025log(40log25log

3

(b)

3

3log3

3log

27log

7

189log7log189log

3

33

3

333

17. Simplify3

3

loglog6

log5

xx

x

, where x > 0 and x 1.

化簡 3

3

loglog6

log5

xx

x

，其中 x > 0 且 x 1。

17

45

3

17

15

log)3

16(

log15

log3

1log6

log15

loglog6

log)53(

loglog6

log5

3

13

3

x

x

xx

x

xx

x

xx

x


9

18. Solve 163 12 x and give your answer correct to 3 significant figures.

解 163 12 x ，並取答案準確至三位有效數字。

fig.) sig. 3 to(cor.76.1

2

13log

16log

3log

16log12

16log3log)12(

16log3log

163

12

12

x

x

x

x

x

19. Given that log 2 = a and log 3 = b, express the following in terms of a and b.

已知 log 2 = a 及 log 3 = b，試以 a 和 b 表示下列各數式。

(a) log 6

(b) log 1800

(a)

ba

3log2log

)32log(6log

(b)

22

10log23log22log

10log3log2log

)1032log(1800log

22

22

ba

20. Solve 06log)(log 2 xx .

解 06log)(log 2 xx 。

Let y = log x, then the equation (log x)2 – log x – 6 = 0 becomes:

0)3)(2(

062

yy

yy

y = –2 or y = 3

For y = –2,

100

1

2log

x

x

For y = 3, log x = 3

x = 1000

∴

1000or100

1x

Question Bank

10

21. Solve 11 35 xx and give your answer correct to 3 significant figures.

解 11 35 xx ，並取答案準確至三位有效數字。


3log5log

5log3log

5log3log)3log5(log

3log3log5log5log

3log)1(5log)1(

3log5log

35

11

11

x

x

xx

xx

xx

xx

22. Solve the following logarithmic equations.

解下列各對數方程。

(a) 2)23log( x

(b) 2log)8log()6log( xx

(a) ∵

2)23log( x

∴

34

1023

10023

x

x

x

(b)

10

1626

28

6

2log8

6log

2log)8log()6log(

x

xx

x

x

x

x

xx

23. Given that the intensity level of the sound produced by a barking dog is 50 dB and that of a crying baby

is 45 dB. How many times is the intensity of the sound produced by the barking dog to that of the crying

baby? (Give your answer correct to 3 significant figures.)

已知一隻狗的吠聲和一名嬰兒的哭聲的聲強級分別是 50 dB 和 45 dB。問狗的吠聲的聲強相當於嬰

兒的哭聲的多少倍？(答案須準確至三位有效數字。)

Let I1 and I2 be the sound intensities produced by the barking dog and the crying baby respectively.

By the definition of sound intensity level, we have

0

1log1050I

I

and

0

2log1045I

I


11

∴


10

log2

1

loglog105

log10log104550

2

1

2

1

2

1

0

2

0

1

0

2

0

1

I

I

I

I

I

I

I

I

I

I

I

I

∴ The sound intensity produced by the barking dog is

3.16 times to that of the crying baby.

24. The strengths of earthquakes in City A and City B were measured 6 and 5.4 respectively on the Richter

scale. How many times is the strength of the earthquake in City A to that in City B? (Give your answer

correct to 3 significant figures.)

已知發生於 A 城和 B 城的地震的強度分別為黎克特制 6 級和 5.4 級。問 A 城的地震的強度是

B 城的多少倍？(答案須準確至三位有效數字。)

Let E1 and E2 be the relative energies released by the earthquakes in City A and City B respectively.

By the definition of the Richter scale, we have

6 = log E1 and 5.4 = log E2

i.e. E1 = 106 and E2 = 105.4

∴

98.3

10

10

10

6.0

4.5

6

2

1

E

E

(cor. to 3 sig. fig.)

∴ The strength of the earthquake in City A is 3.98 times that in City B.

25. The sound intensity inside a concert hall is 3 times as that outside the concert hall. If the sound intensity

level outside the concert hall is 45 dB, find the sound intensity level inside the concert hall. (Give your

answer correct to 3 significant figures.)

已知演奏廳內的聲強是演奏廳外的 3 倍。若演奏廳外的聲強級是 45 dB，求演奏廳內的聲強級。(答

案須準確至三位有效數字。)

Let I be the sound intensity outside the concert hall, then the sound intensity inside the concert hall is 3I.

∵ The sound intensity level outside the concert hall is

45 dB.

∴

0

log1045I

I

Let dB be the sound intensity level inside the concert hall.

∴

0

3log10

I

I


3log1045

3log1045

log3

loglog1045

log103

log1045

00

00

I

I

I

I

I

I

I

I

∴ The sound intensity level inside the concert hall is

Question Bank

12

49.8 dB.

26. Solve

x

xx

3

12

8

1644

.

解

x

xx

3

12

8

1644

。

x

xx

3

12

8

1644

xxx 331622 )2()2()2(

xxx 9664 222

6

9610

22

22

9610

9664

x

xx

xx

xxx

27. Simplify the following expression and express your answer with positive indices.

化簡下列各題，並以正指數表示答案。

37

33

1

3

2

1

827

9

1

a

b

b

a

37

33

1

3

2

1

827

9

1

a

b

b

a

=

3

17

3

13

3

1

3

1

2

13

3

1

3

1

2

11 827

9

a

b

b

a

=

3

7

2

1

3

1

2

123

9

a

b

b

a

= 2

11

3

7

3

1

63

ba

= 2

1

218 ba

=2

2

1

18

a

b


13

28. Solve .12log)1(log6 23

1

2 xx

解 12log)1(log6 23

1

2 xx 。

1

0)1(

012

412

4)1(

22

)1(

12

)1(log

12log)1(log

12log)1(log

12log)1(log6

2

2

2

2

2

2

2

22

2

2

63

1

2

23

1

2

x

x

xx

xxx

xx

x

x

x

x

xx

xx

xx

29. Simplify 2log3log

27

8log

2

3log3

22

2

2

2

.

化簡 2log3log

27

8log

2

3log3

22

2

2

2

。

9

2

3log

2

3log)36(

2

3log

2

3log3

2

3log6

2

3log

2

3log

2

3log6

2

3log

3

2log

2

3log)23(

2log3log

27

8log

2

3log3

2

2

2

22

2

3

22

2

3

22

22

2

2

2

Question Bank

14

Pre-requisite Questions

預備測驗

1. Find the value of 543 555 .

求 543 555 的值。

25

5

5555

2

543543

2. Find the value of 99 kk

.

求 99 kk

的值。

1

0

9999

k

kkk

3. Find the value of 32 )3( .

求 32 )3( 的值。

729

3

3)3(

6

3232


求 23 22 的值。

32

2

222

5

2323


求 36 44 的值。

64

4

444

3

)3(636

6. Find the value of 4

0

27

1

.

求 4

0

27

1

的值。


15

16

2

2

1

2127

1

4

4

440


10

2

45

.

求 2

10

2

45

的值。

3

4

14

14

2

14

11

2

45

2

2

10

8. Find the value of

202

7

4

8

7

7

6

.

求

202

7

4

8

7

7

6

的值。

9

4

3

2

6

4

7

4

6

7

7

41

6

7

7

4

8

7

7

6

2

2

2

2

2

2

22

202

9. Find the value of 102 )33( .

求 102 )33( 的值。

9

3

)3(

)13()33(

2

12

12102

Question Bank

16


3

12

3

.

求

3

12

3

的值。

216

1

)6(

23

2

3

3

3

3

1

11. Simplify and express 527 aaa .

化簡 527 aaa 。

10

527527

a

aaaa

12. Simplify and express 23 )4( a with positive indices.

化簡 23 )4( a ，並以正指數表示答案。

6

23223

16

)4()4(

a

aa

13. Simplify and express 84 bb with positive indices.

化簡 84 bb ，並以正指數表示答案。

12

84

)8(484

b

b

bbb

14. Simplify and express 5202 )2()3( nmnm with positive indices.

化簡 5202 )2()3( nmnm ，並以正指數表示答案。

10

5

5525

525202

32

2

)2(1)2()3(

m

n

nm

nmnmnm

15. Simplify and express

61

bwith positive indices.

化簡

61

b，並以正指數表示答案。


17

6

616

)(1

b

bb

16. Simplify and express 312 )( ba with positive indices.

化簡 312 )( ba ，並以正指數表示答案。

6

3

36

)3(1)3(2312 )(

a

b

ba

baba


11

1

y

x

y

xwith positive indices.

化簡

11

1

y

x

y

x，並以正指數表示答案。

xy

xyxy

y

xxy

y

x

y

x

2

1

11

1


4

3

20

t

srwith positive indices.

化簡

4

3

20

t

sr，並以正指數表示答案。

128

)4(3)4(2

432

4

3

24

3

20

)(

1

ts

ts

ts

t

s

t

sr

Question Bank

18

Level 1 Questions

程度 1 題目

1. Simplify and express 22

43

)9(

)3(

a

awith positive indices.

化簡 22

43

)9(

)3(

a

a，並以正指數表示答案。

8

412

4

12

222

434

22

43

81

81

)(9

)(3

)9(

)3(

a

a

a

a

a

a

a

a


2

52

651

6

18

a

ba with positive indices.

化簡

2

52

651

6

18

a

ba，並以正指數表示答案。

12

20

1220

2610

2655

2

52

651

4

4

)2(

18

36

6

18

b

a

ba

ba

baa

ba


61

)4(

)2(

b

b with positive indices.

化簡 441

61

)4(

)2(

b


4

4

)(256

)(64

)()4(

)(2

)4(

)2(

10

)16(6

16

6

4441

616

441

61

b

b

b

b

b

b

b

b


19


3

35

22

ba


化簡

3

35

22

ba


15

21

1521

357

332)5(2

3

35

22

)(

][

b

a

ba

ba

baba

ba


243

1

.

求 5

243

1

的值。

3

1

3

1

243

1 5

5

5


求 6 64 的值。

2

2646 66


1

729

.

求 3

1

729

的值。

9

1

)9(

])9[(

)729(729

1

13

13

13

1

3

1

Question Bank

20


1

256 .

求 4

1

256 的值。

4

)4(256 4

1

44

1

9. Express 5 2x in the form xp, where p is a rational number.

把 5 2x 寫成 xp 的形式，其中 p 是一個有理數。

5

2

5

125 2 )(

x

xx

10. Express 63 )( x in the form xp, where p is a rational number.

把 63 )( x 寫成 xp 的形式，其中 p 是一個有理數。

2

63

1

63 )()(

x

xx

11. Express 4 681

9

x

x in the form x

p, where p is a rational number.

把 4 681

9

x

x 寫成 x

p 的形式，其中 p 是一個有理數。

2

1

2

31

2

3

4

164

14 6

3

3

3

9

)()81(

9

81

9

x

x

x

x

x

x

x

x

12. Express 3)(

2

x in the form x

p, where p is a rational number.

把 3)(

2

x 寫成 x

p 的形式，其中 p 是一個有理數。


21

2

3

2

3

32

13

2

2

)(

2

)(

2

x

x

xx


2

)216( without using a calculator.

試不使用計算機，求 3

2

)216( 的值。

36

)6(

])6[()216(

2

3

233

2


4

125

64

without using a calculator.


4

125

64

的值。

625

256

5

4

5

4

125

64

4

3

43

3

4


3

16

625



3

16

625

的值。

8

125

2

5

2

5

16

625

3

4

34

4

3

Question Bank

22

16. Simplify and express 3 2

4 3

a

a with positive indices.

化簡 3 2

4 3

a


12

1

12

89

3

2

4

3

3

2

4

3

3

12

4

13

3 2

4 3

)(

)(

a

a

a

a

a

a

a

a

a


3

49

36



3

49

36

的值。

216

243

6

7

6

7

36

49

49

36

3

2

32

2

3

2

3


5

2

1

3

2

43 )()8(

baba with positive indices.

化簡 46

5

2

1

3

2

43 )()8(

baba ，並以正指數表示答案。

4

4

)2(

)()8()()8(

3

2

3

20

3

23

3

10

3

8

22

3

1023

823

246

5

2

1

3

243

b

ba

ba

babababa


23


4

)2(

144a


化簡 31

4

)2(

144a


5

)3(2

3

2

33

22

31

4

2

3

2

3

8

12

2

)12(

)2(

144

a

a

a

a

a

a

a

a


1

9652 )()( baab with positive indices.

化簡 3

1

9652 )()( baab ，並以正指數表示答案。

7

7

77

31025

321053

19652 )()()()(

b

a

ba

ba

bababaab


2

4

1

6

5

25

4

3

1

)32()3(

baba with positive indices.

化簡 5

2

4

1

6

5

25

4

3

1

)32()3(

baba ，並以正指數表示答案。

2

3

3

1

2

3

3

1

2

10

1

5

8

3

1

3

2

10

1

3

1

5

25

5

8

3

2

10

1

3

1

5

2

5

8

3

22

5

2

4

1

6

525

4

3

1

36

36

)2(

9

])2[(

9

])32[()3(

)32()3(

ba

ba

ba

ba

ba

baba

baba

Question Bank

24


4

3

2

3

2

2

256

81

b


化簡

4

3

2

3

2

2

256

81

b


2

2

3

23

2

3

3

24

3

4

2

3

4

3

4

4

3

3

8

4

3

4

3

24

3

4

3

3

8

2

4

3

2

3

8

4

3

2

3

22

4

3

3

2

2

24

3

2

3

2

2

27

64

3

4

)3(

)4(

)()81(

)()256(

81

256

81

256

81

256

81

256

256

81

b

a

b

a

b

a

b

a

b

a

a

b

a

b

ba

b

b

ba

23. Solve the equation 4

31

3

2

2564

x

.

解 4

31

3

2

2564

x

。


25

∴ 6

313

2

44

)4(4

2564

31

3

2

4

34

13

2

4

31

3

2

x

x

x

x

x

24 Solve the equation 5

2

2 2433

x

.

. 解 5

2

2 2433 x

。

4

22

33

)3(3

2433

22

5

252

5

2

2

x

x

x

x

x

25. Solve the equation 72)3(33 3 xx .

解 72)3(33 3 xx 。

∴ 1

33

72)24(3

72)33(3

72)3(33

3

3

x

x

x

x

xx


2

4 279

x

.

解 3

2

4 279 x

。

∴ 4

22

33

)3()3(

279

22

3

2342

3

2

4

x

x

x

x

x

Question Bank

26


4)4(4 1

xx .

解 24016

4)4(4 1

xx 。

4

22

44

164

240)14(4

24044

24016

4)4(4

22

2

22

2

1

x

x

x

x

x

xx

xx

28. Solve the equation )3(8193 xx .

解 )3(8193 xx 。

5

4

46

33

)3)(3()3(

)3(819

46

432

3

x

xx

xx

xx

xx

29. Solve the equation 12642 142 xx .

解 12642 142 xx 。

2

3

1)1(2

22

24

126)14(4

12644

12642

12642

)1(2

1

31

12

1)2(2

142

x

x

x

x

x

xx

xx

xx

30. Solve the equation )9(2735 xx .

解 )9(2735 xx 。

1

235

33

)3)(3(3

)9(273

235

235

5

x

xx

xx

xx

xx


27

31. Find the value of log 1000 without using a calculator.

試不使用計算機，求 log 1000 的值。

3

10 log1000 log 3

32. Find the value of 000 100

1 log without using a calculator.

試不使用計算機，求 000 100

1 log 的值。

5

10 log000 100

1log 5

33. Find the value of log 0.000 1 without using a calculator.

試不使用計算機，求 log 0.000 1 的值。

4

10 log1 0.000 log 4


1 log

3 without using a calculator.


1 log3 的值。

3

3log

3

1log

729

1log

3

3

633

35. Find the value of 128 log2


試不使用計算機，求 128 log2 的值。

7

2log128log 7

22

36. Find the value of 0.008 log5


. 試不使用計算機，求 0.008 log5 的值。

3

5log

125

1log

1000

8log008.0log

3

5

5

55

Question Bank

28

37. Find the value of x in the equation 710 x and give your answer correct to 3 significant figures.

解 710 x ，並取答案準確至三位有效數字。

∵ 10x = 7

∴ x = log 7

= 845.0 (cor. to 3 sig. fig)



∵ 4010 x

∴

fig.) sig. 3 to(cor. 60.1

40 log

x



∵ 80103 x

∴


80 log3

x



∵ 16910 x

∴


169 log

x

41. Find the value of x in the equation 5.010 x and give your answer correct to 3 significant figures.

解 5.010 x ，並取答案準確至三位有效數字。

∵ 5.010 x

∴


0.5 log

x

42. Find the value of log 25 + log 400 without using a calculator.

試不使用計算機，求 log 25 + log 400 的值。

4

10 log

000 10 log

)40025log(400 log25 log

4


29

43. Find the value of x in the equation 2.9)10( x and give your answer correct to 3 significant

figures.

解 2.9)10( x ，並取答案準確至三位有效數字。

x)10( = 9.2

210

x

= 9.2

2

x = log 9.2

x = 93.1 (cor. to 3 sig. fig.)

44. Find the value of log 8 + log 625 log 5 without using a calculator.

試不使用計算機，求 log 8 + log 625 log 5 的值。

3

10 log

1000 log

5

6258log5 log625 log8 log

3

45. Find the value of 32log2log44


試不使用計算機，求 32log2log44

的值。

1

4log

4

1log

16

1log

32

2log32log2log

1

4

4

4

444

46.. Find the value of

64

1log

16log

6


試不使用計算機，求

64

1log

16log

6

3

6 的值。

Question Bank

30

9

2

2log 6

2log3

4

2log

2log

2log

)2(log

64

1log

16log

6

6

66

3

4

6

66

3

1

46

6

36

47. Find the value of 25log

125log

5


試不使用計算機，求 25log

125log

5

5 的值。

2

3

5log 2

5log 3

5log

5log

25log

125log

5

5

2

5

3

5

5

5

48. Find the value of 243log

81log without using a calculator.

試不使用計算機，求 243log

81log 的值。

5

4

3 log 5

3 log 4

3 log

3 log

243 log

81 log5

4

49. Given that log 3 = x and log 4 = y, express log 12 in terms of x and y.

已知 log 3 = x 及 log 4 = y，試以 x 和 y 表示 log 12。

yx

4 log3 log

4)log(312 log


31



yx 23

4 log 2 3 log 3

4 log3 log

)4log(3432 log

23

23



xy

13

2 log13 log 3

2 log10 log3 log

2

103log

)5log(3135 log

3

3

3

52. Solve the equation 83 1 x and give your answer correct to 3 significant figures.

解 83 1 x ，並取答案準確至三位有效數字。


13 log

8 log

8log3log)1(

8 log3 log

83

1

1

x

x

x

x

53 Solve the equation 42)2(3)2(2 21 xx and give your answer correct to 3 significant figures.

解 42)2(3)2(2 21 xx 。並取答案準確至三位有效數字。


22 log

6 log

6 log2 log)2(

62

42)32(2

42)2(32

42)2(3)2(2

2

22

2

21

x

x

x

x

xx

xx



13

10 log 3 log2 log 3

10 log3 log2 log

)103log(2240 log

3

yx

Question Bank

32

55 Solve the equation 123 45 xx and give your answer correct to 3 significant figures.



5 log 34 log 2

4 log

4 log5) log 34 log (2

4 log4 log 25 log 3

4log)12(5 log 3

45 123

x

x

xx

xx

xx

56. Solve the equation 452 21 xx and give your answer correct to 3 significant figures.



2 log5 log 2

2 log 3

2 log 32) log 5 log (2

2 log )3(5 log 2

25

25

452

32

)1(22

21

x

x

xx

xx

xx

xx

57. Solve the equation 1)15(log4 x .

解 1)15(log4 x 。

4

115

4

1log)15(log

1)15(log

44

4

x

x

x

∴ 4

1x

58. Solve the equation log (3x 2) = 2.

解 log (3x 2) = 2。

10023

100 log)2log(3

22)log(3

x

x

x

∴ 34x


33


解 5)1(log2 x 。

321

32log)1(log

5)1(log

22

2

x

x

x

∴ 31x


解 0)54(log7 x 。

154

0)54(log7

x

x

∴ 1x

61. The sound intensity level of piledriver A and piledriver B are 108 dB and 114 dB respectively. How

many times is the sound intensity produced by piledriver B to that by piledriver A?

(Give your answer correct to 3 significant figures.)

打椿機 A 和打椿機 B 所產生的聲強級分別是 108 dB 和 114 dB。問打椿機 B 所產生的聲強

是打椿機 A 的多少倍?

（答案須準確至三位有效數字。）

Let I1 and I2 be the sound intensities produced by piledriver A and piledriver B respectively.


and

0

2

0

1

log 10114

log 10108

I

I

I

I

∴


10

log5

3

loglog106

log 10log 10108114

5

3

1

2

1

2

0

1

0

2

0

1

0

2

I

I

I

I

I

I

I

I

I

I

I

I

∴ The sound intensity produced by piledriver B is

3.98 times to that of piledriver A.

Question Bank

34

62. The sound intensity level of a road is 75 dB measured from the building next to the road. After

building noise absorber along the road, the sound intensity level decreases to 65 dB. How many times

is the sound intensity of the road before building noise absorber to that after building noise absorber?

一條街道的路面噪音的聲強級是 75 dB。當設置吸音裝置後，路面噪音的聲強級減少至 65 dB。

問設置吸音裝置前的路面噪音的聲強是設置吸音裝置後的多少倍？

Let I1 and I2 be the sound intensities measured on the road before and after building noise absorber respectively.


and

0

2

0

1

log 1065

log 1075

I

I

I

I

∴

10

10

log1

loglog1010

log 10log 106575

1

2

1

2

1

0

2

0

1

0

2

0

1

I

I

I

I

I

I

I

I

I

I

I

I

∴ The sound intensity measured on the road before building noise absorber is 10 times to that after building noise absorber.

63. The strengths of earthquakes in city A in 1979 and in 1995 were measured 6.5 and 7.2 respectively on

the Richter scale. How many times is the strength of the earthquake in city A in 1995 to that in 1979?


城市 A 於 1979 年和 1995 年的地震強度分別是黎克特制 6.5 級和 7.2 級。問城市 A 於

1995 年的地震強度是 1979 年的多少倍？


Let E1 and E2 be the relative energies released by the earthquakes in city A in 1979 and in 1995 respectively.


6.5 log E1 and 7.2 log E2

i.e. E1 106.5 and E2 107.2

∴


10

10

10

7.0

5.6

2.7

1

2

E

E

∴ The strength of the earthquake in city A in 1995 is 5.01 times to that in 1979.


35

64. The sound intensity level of a construction site in the evening is 50 dB, whereas the sound intensity

level during the day-time increases by 30 dB. How many times is the sound intensity of the

construction site during the day-time to that in the evening?

一個建築工程地盤在黃昏時所產生的聲強級是 50 dB，而在日間所產生的聲強級較黃昏時的增

加 30 dB。問該建築工程地盤在日間時的聲強是黃昏時的多少倍？

Let I1 and I2 be the sound intensities of the construction site in the evening and the day-time respectively.


1000

10

3log

30loglog10

30log 10log 10

3

1

2

1

2

0

1

0

2

0

1

0

2

I

I

I

I

I

I

I

I

I

I

I

I

∴ The sound intensity of the construction site during the day-time is 1000 times to that in the evening.

65. How many times is the strength of an earthquake measured 7 on the Richter scale to that measured 4

on the Richter scale?

問黎克特制 7 級地震的強度是黎克特制 4 級地震的多少倍?

Let E1 and E2 be the relative energies released by the earthquakes measured 7 and 4 on the Richter scale respectively.


7 log E1 and 4 log E2

i.e. E1 107 and E2 104

∴

1000

10

10

10

3

4

7

2

1

E

E

∴ The strength of the earthquake measured 7 on the Richter scale is 1000 times to that measured 4 on the Richter scale.

Question Bank

36

Level 2 Questions

程度 2 題目


5

2

6 3

x

xx with positive indices.

化簡 3

5

2

6 3

x

xx ，並以正指數表示答案。

5

3

2

3

5

2

2

1

2

3

5

2

2

1

2

3

5

2

6

3

3

5

2

6 3

1

x

x

x

xx

x

xx

x

xx


1

2

1

6

1

3

2

4

1

)(mn

nm

nm with positive indices.

化簡 3

1

2

1

6

1

3

2

4

1

)(mn

nm

nm ，並以正指數表示答案。

6

1

4

1

6

1

4

1

3

1

2

1

3

2

3

1

6

1

4

1

3

1

3

1

2

1

6

1

3

2

4

1

3

1

2

1

6

1

3

2

4

1

1

)(

nm

nm

nm

nm

nm

nmmn

nm

nm

3. Simplify and express )6(272 4

3

3 26

1

1

yyy with positive indices.

化簡 )6(272 4

3

3 26

1

1

yyy ，並以正指數表示答案。


37

4

5

4

5

3

2

4

3

6

1

3

2

4

3

6

1

4

3

3

1

26

1

14

3

3 26

1

1

1

3

3

)6()27(2)6(272

y

y

y

y

y

yyyyyy

4. Simplify and express )()( 4

1

3

1

22

3

4 2

abbaba with positive indices.

化簡 )()( 4

1

3

1

22

3

4 2

abbaba ，並以正指數表示答案。

3

2

2

4

1

3

2

4

11

2

1

2

1

4

1

3

2

2

1

4

1

2

1

4

1

3

2

2

1

4

1

24

1

3

1

22

3

4 2

)()()(

)()()()()(

ba

ba

abbaba

abbabaabbaba

5 Simplify and express 23

2

3

13

2

3 )4()3(64

27babaab

with positive indices.

化簡 23

2

3

13

2

3 )4()3(64

27babaab

，並以正指數表示答案。

3

3

2123

4

3

1

3

2

23

4

3

1

23

2

23

4

23

1

23

23

2

23

2

3

13

2

3

3

3

1

3

1

)16()3(9

16

)4()3(64

27

)4()3(64

27

b

a

ab

ba

bababa

bababa

babaab

Question Bank

38


1

2

4

3

)(

b

aba

b


化簡

1

2

4

3

)(

b

aba

b


6

60

)1(1)4(123

1

12

4

312

4

3

)()(

b

ba

ba

b

aba

b

a

b

aba

b

a


3

)()5(25 xyyxx

with positive indices.

化簡 4324

3

)()5(25 xyyxx

，並以正指數表示答案。

3

4)2(

3

4

2

3

2

1

)2(1

3

4

3

4

22

3

22

1

3

4

3

4

22

3

22

1

424

3

5

)()5()5(

)()5()25(

)()5(25 3

yx

yxyxx

yxyxx

xyyxx

3

10

3

1

3

10

3

1

3

125

5

yx

yx


39

8. Simplify and express 13 634

1

4 24)8(

2 ababa

b with positive indices.

化簡 13 634

1

4 24)8(

2 ababa


b

a

ba

ba

aba

ab

b

aba

ba

b

aba

ba

b

8

7

18

7

2

3

2

11)2(

8

11

22

3

8

1

2

1

3

6

3

1

2

1

34

1

4

1

24

13 634

1

4 24

4

4

4

2)(

2

)8()(

)(

2

)8(2


25

3

2

4 82

32

3 81

yx

xy

xyx with positive indices.

化簡 5

2

5

3

2

4 82

32

3 81

yx

xy

xyx ，並以正指數表示答案。

5

13

5

130

5

2)1(2)2(

2

33

2

1

5

22

12

3

322

1

5

25

3

3

2

4 82

9

4

9

4

9

4

427

)3(

32

381

y

yx

yx

yx

yx

xyx

yx

xy

xyx

Question Bank

40

10. Simplify and express )2(1

)16( 34

14

3

3 2

2

1

4

nm

mnnm with positive indices.

化簡 )2(1

)16( 34

14

3

3 2

2

1

4

nm

mnnm ，並以正指數表示答案。

32

3

)3(2

1

2

1

4

1

4

12

34

1

2

1

4

1

2

1

2

34

1

2

1

4

12

1

2

34

14

3

3

2

3

12

1

2

34

14

3

3 2

2

1

4

2

2

)2()()4(

)2(1

)4(

)2(1

)4(

)2(1

)16(

nm

nm

nmnmnm

nm

nm

nm

nm

nm

nm

nm

mn

nm

11 Solve the equation 09)3(832 xx .

解 09)3(832 xx 。

(rejected) 93or 13

0)93)(13(

09)3(8)3(

09)3(83

2

2

xx

xx

xx

xx

∴ 0x

12. Solve the equation 40222 112 xxx .

解 40222 112 xxx 。

31

22

82

40)5(2

40)122(2

40222

31

1

1

231

112

x

x

x

x

x

xxx

∴ 4x


41

13. Solve the equation 345)5(3)5(2 11 xxx .

解 345)5(3)5(2 11 xxx 。

01

15

34)15352(5

345)5(3)5(2

1

21

11

x

x

x

xxx

∴ 1x

14. Solve the equation 04)4(17)4(4 2 xx .

解 04)4(17)4(4 2 xx 。

0)44](1)4(4[

04)4(17)4(4

04)4(17)4(4

2

2

xx

xx

xx

∴ 1or 1

44or 44 11

x

xx

15. Solve the simultaneous equations

813

2552

23

yx

yx

.

解聯立方程

813

2552

23

yx

yx

。

813

255

2

23

yx

yx

)2(

)1(

From (1),

∴ 223

55 223

yx

yx

From (2),

∴ 42

33 42

yx

yx


42

223

yx

yx

)4(

)3(

2 (4) (3),

6

6)23()24(

2)4(2)23()2(2

x

yxyx

yxyx


8

4)6(2

y

y

Question Bank

42



110

21662 yx

yx

.

解聯立方程

110

21662 yx

yx

。

110

2166

2 yx

yx

)2(

)1(

From (1),

∴ 3

66 3

yx

yx

From (2),

∴ 02

1102

yx

yx


02

3

yx

yx

)4(

)3(

(3) + (4),

1

33

03)2()(

x

x

yxyx


2

31

y

y



0)8(24

16

12 3

yx

yx

.

解聯立方程

0)8(24

16

12 3

yx

yx

。

......(2) 0)8(24

......(1) 16

12 3

yx

yx

From (1),

∴ 43

22 43

yx

yx

From (2),


43

∴ 132

132

22

)2(22

132

32

yx

yx

yx

yx


132

43

yx

yx

)4(

)3(

(3) + (4),

1

33

14)32()3(

x

x

yxyx


1

33

43)1(

y

y

y



64)2(4

033 12

yx

yx

.

解聯立方程

64)2(4

033 12

yx

yx

。

64)2(4

033 12

yx

yx

)2(

)1(

From (1),

∴ 12

12

33 12

xy

yx

yx

From (2),

∴ 4

62

22 62

yx

yx

yx


4

12

yx

xy

)4(

)3(

By substituting (3) into (4), we have

1

413

4)12(

x

x

xx

Question Bank

44


3

1)1(2

y



125

1log2log3 without using a calculator.


125

1log2log3 的值。

3

1000 log

125)log(8

125

18log

125

1log2 log

125

1log2 log 3 3


3log212log27log

3

1222



3log212log27log

3

1222

的值。

6

2log

64log

16

9

123log

16

9log12log3log

4

3log12log27log

4

3log 212log27log

3

1

6

2

2

2

222

2

22

3

2

222


2log29

1log

36log4

1216log 3

aa

aa



2log29

1log

36log4

1216log 3

aa

aa

的值。


45

4

3

22

3

)6(log

)(6log

6

1log

)6(6log

4log9

1log

36log6log

2log 29

1log

36log4

1216log

2

2

3

2

2

1

4

3

a

a

a

a

aa

aa

aa

aa


150

1log18log

2

1

9log5.227log2



150

1log18log

2

1

9log5.227log2

的值。

1

3log

3log

110log3log

3log

1)103log(

3log

1)25log()23log(

3

1log

1)50log()18log(

27

3log

150

1log)18log(

27log)9log(

150

1log18log

2

1

9log5.227log2

2

5

22

5

Question Bank

46

23. Simplify 3

332

log

loglog3

1

x

yxyx

, where x > 0, y > 0 and x, y 1.

化簡 3

332

log

loglog3

1

x

yxyx

，其中 x > 0，y > 0，且 x，y 1。

7

log3

1

log 3

7

log

)log(

log

)log(

log

log)log(

log

log)log(

log

log log 3

1

3

1

3

7

3

1

33

2

3

1

33

2

3

1

33

1

32

3

332

x

x

x

x

x

x

x

yxyx

x

yxyx

x

yxyx

24. Simplify

yx

x

2

13

327

9log , where x > 0, y > 0 and x, y 1.

化簡

yx

x

2

13

327

9log ，其中 x > 0，y > 0，且 x，y 1。

)32(2

1

3

3log

2

1

3

3log

2

1

27

9log

2

1

27

9log

36

26

3

)2(3

)13(2

3

2

13

32

13

3

y

yx

x

yx

x

yx

x

yx

x


47

25. Simplify 3

4

loglog2

log6log3

yy

x

yx

, where x > 0, y > 0 and x, y 1.

化簡 3

4

loglog2

log6log3

yy

x

yx

，其中 x > 0，y > 0，且 x，y 1。

6

)log(

)log( 6

)(log

)log(

)log(

)(log

loglog

log log

loglog 2

6log log 3

2

2

2

62

2

612

3

2

2

612

3

4

yx

yx

yx

yx

yx

yx

yy

x

yx

yy

x

yx

26. Simplify

yxxy

xyxy

2

3

loglog2

3

loglog

, where x > 0, y > 0 and x, y 1.

化簡

yxxy

xyxy

2

3

loglog2

3

loglog

，其中 x > 0，y > 0，且 x，y 1。

1

)log(2

1

)log(2

1

log

log

)log(

log

log)log(

log)log(

log log2

3

loglog

1

1

2

1

2

1

2

1

2

1

22

3

2

3

2

3

2

1

22

3

2

3

2

3

2

1

2

3

xy

xy

yx

yx

yxyx

xy

yx

yxyx

xyyx

yxxy

xyxy

Question Bank

48



12

15 log 26 log

10 log5 log6 log

10

56log

10

150log15 log

2

2

yx



xy

4 log24 log

4

24log6 log

29. Solve the equation 24log)12log( x .

解 24log)12log( x 。

2512

100)14(2

100 log)]1log[4(2

24 log)12log(

x

x

x

x

∴ 13x



xy2

3

4log2

324log

2log2

14log24 log

2log4log24log

24

24log3 log

2

31. Given that log 5 = x and log 6 = y, express 288log in terms of x and y.

已知 log 5 = x 及 log 6 = y，試以 x 和 y 表示 288log 。


49

)33(2 2

1

5) log 310log36 log 2(2

1

)5 log10 log6 log(2

1

5

106log

2

1

)2log(62

1

288 log2

1288log

332

3

32

32

xy

32. Solve the equation 34log3)23(log 22 x .

解 34log3)23(log 22 x 。

8

1

64

23

8

1log

64

23log

34log)23(log

34log 3)23(log

22

322

22

x

x

x

x

∴ 2x

33. Solve the equation 04)34(log3)]34([log 3

2

3 xx .

解 04)34(log3)]34([log 3

2

3 xx 。

0]4)34(log][1)34(log[

04)34(log 3)]34(log[

33

3

2

3

xx

xx

8134 or

3

134

4)34(logor 1)34(log 33

xx

xx

∴ 21 or 6

5 xx

34 Solve the equation )1010log(3

2)4log()1log( xx .

解 )1010log(3

2)4log()1log( xx 。

Question Bank

50

(rejected) 6or 1

0)6)(1(

065

10)4)(1(

1)]4)(1log[(

)log(103

24)]1)(log[(

)10log(103

2)4log(1)log(

2

2

3

xx

xx

xx

xx

xx

xx

xx

∴ 1x


1log2log

82

yx

yx.

解聯立方程

1log2log

82

yx

yx。

1 log 2 log

82

yx

yx

)2(

)1(

From (2),

2

2

2

10

)log(10 log

1 log log

yx

yx

yx

……(3)


(rejected) 1or 5

4

0)1)(45(

045

45

82)10(

2

2

2

yy

yy

yy

yy

yy

By substituting 5

4y into (1), we have

5

32

85

42

x

x

∴ The solution is 5

32x ,

5

4y .


18log2)2log(

17

3

3

yx

yx

.

解聯立方程

18log2)2log(

17

3

3

yx

yx

。


51

18log2)log(2

17

3

3

yx

yx

)2(

)1(

From (1),

03 yx

∴ xy 3

From (2),

104

2

14

2log

12 log 2)2log(

yx

yx

yx

∴ 402 yx


402

3

yx

xy

)4(

)3(


8

405

4032

x

x

xx


24

)8(3

y



3log1loglog2

0)log(

444 yx

yx.

解聯立方程

3log1loglog2

0)log(

444 yx

yx。

3log1loglog 2

0)log(

444 yx

yx

)2(

)1(

From (1),

yx

yx

1

1

From (2),

)34(log)(log

3log4log)(log

3log1loglog

44

444

444

xy

xy

yx

Question Bank

52

∴ 12xy


12

1

xy

yx

)4(

)3(


(rejected) 4or 3

0)4)(3(

012

12)1(

2

yy

yy

yy

yy

By substituting y = 3 into (3), we have

4

)3(1

x



yxy

yx

3

4

3

126

log32log

03)9(3.

解聯立方程

yxy

yx

3

4

3

126

log32log

03)9(3。

yxy

yx

34

3

126

log 32log

03)9(3

)2(

)1(

From (1),

12213

33

3]3[3

3)9(3

12213

12)6(2

126

yx

yx

yx

yx

∴ xy 6

From (2),

9

9loglog

2loglog

33

4

3

3

3

4

3

xy

y

xy

yxy


9

6

xy

xy

)4(

)3(



53

3

0)3(

096

9)(6

2

2

x

x

xx

xx


3

)3(6

y


39. The sound intensity level of a busy road is 80 dB. During traffic jam, the sound intensity of traffic on

the road is reduced to 0.1% of the original. Find the sound intensity level of the road during traffic

jam.

一條繁忙的街道的路面噪音的聲強級是 80 dB。當交通擠塞時，路面所產生的聲強比平時減少

0.1%。求交通擠塞時該條街道的路面噪音的聲強級。

Let I be the sound intensity before traffic jam, then the sound intensity during traffic jam is I × 103.

∵ The sound intensity level before traffic jam is

80 dB.

∴

0

log 1080I

I

Let dB be the sound intensity level of the road during traffic jam.

∴

0

310log 10

I

I

50

31080

)log(10 1080

10loglog1080

10log 10log 1080

3

0

3

0

0

3

0

I

I

I

I

I

I

I

I

∴ The sound intensity level of the road during traffic jam is 50 dB.

40. The sound intensity level in a gathering is 45 dB. After playing music, the sound intensity doubles.

What is the sound intensity level after playing music?


一次聚會開始時所產生的聲強級是 45 dB。當音樂播放後，聲強高了一倍。求播放音樂後所產

生的聲強級。


Let I be the sound intensity before playing music, then the sound intensity after playing music is 2I.

Question Bank

54

∵ The sound intensity level before playing music is

45 dB.

∴

0

log 1045I

I

Let dB be the sound intensity level after playing music.

∴

0

2log 10

I

I


2 log 1045

2 log 1045

log2

log1045

log 102

log 1045

00

00

I

I

I

I

I

I

I

I

∴ The sound intensity level after playing music is

48.0 dB.

41. If the strength of an earthquake is

5

1times to that measured 8.8 on the Richter scale, what is the

magnitude of the required earthquake on the Richter scale?


若一地震的強度是黎克特制 8.8 級的 5

1 倍，那麼該地震的強度相等於黎克特制多少級呢？


Let E be the relative energy released by the earthquake measured 8.8 on the Richter scale.

Let R be the magnitude of the required earthquake on the Richter scale.


8.85

1log

log5

1log

5

1log

E

ER

∴ The magnitude of the required earthquake is 8.10 on the Richter scale.

42 If the sound intensity in a construction site is increased by 10%, what is the increase of the

corresponding sound intensity level?


若一個建築工程地盤的聲強增加 10%，則相應的聲強級會增加多少？


Let I be the initial sound intensity, then the sound intensity after the increase is 1.1I.


55

∴ The increase in the corresponding sound intensity level

fig.) sig. 3 to(cor. dB 0.414

dB )1.1 log 10(

dB log1.1

log10

dB log 101.1

log 10

00

00

I

I

I

I

I

I

I

I

43. If the strength of an earthquake is 10 times to that measured 4.3 on the Richter scale, what is the

magnitude of the required earthquake on the Richter scale?

若一地震的強度是黎克特制 4.3 級的 10 倍，那麼該地震的強度相等於黎克特制多少級呢？

Let E be the relative energy released by the earthquake measured 4.3 on the Richter scale.

∴ 4.3 = log E

Let R be the magnitude of the required earthquake on the Richter scale.

∴

3.5

3.41

log10 log

)log(10

E

ER

∴ The magnitude of the required earthquake is 5.3 on the Richter scale.

Question Bank

56

Level 2+ Questions

程度 2+ 題目

1 (a) It is given that f(x) = x4 5x

3 + 20x 16.

已知 f(x) = x4 5x

3 + 20x 16。

(i) Find the values of f(4), f(6) and f(10).

求 f(4)、f(6) 和 f(10) 的值。

(ii) Hence, factorize f(x) completely.

由此，因式分解 f(x)。

(b) Given that log 2 = a and log 3 = b. By using the above results, express the following in terms of

a and b.

已知 log 2 = a 及 log 3 = b。試利用以上的結果，以 a 和 b 表示下列各對數的值。

(i) log 3200 (ii) log 518.4

(a) (i)

0

1680320256

16)4(20)4(54)4( 34

f

320

1612010801296

16)6(20)6(56)6( 34

f

5184

16200500000010

16)10(20)10(510)10( 34

f

(ii) ∵ From (a)(i), we have f(4) = 0.

∴ x – 4 is a factor of f(x).

By long division,

)2)(2)(1)(4(

)2)(1)(4(

)]1(4)1()[4(

)44)(4(

16205

22

2

23

34

xxxx

xxx

xxxx

xxxx

xxx

∴ )2)(2)(1)(4()( xxxxxf

(b) (i)

25

22 l o g 5

22 l o g

1)2( 1 0 l o g

1)225(2 log

1)]26)(26)(16)(4[(6 log

1320 log

10)(320 log3200 log

5

5

23

a


57

(ii)

146

13 log 42 log 6

13 log2 log

1)3(2 log

1]2)32(33)[(2 log

18)129(6 log

12)](10

2)1)(104)(10[(10 log

15184 log

10

5184 log518.4 log

46

46

322

ba

2 Given that 32z

xy. If k

zyx

2

log

2

log

1

log222 , find

已知 32z

xy。若 k

zyx

2

log

2

log

1

log 222 ，求

(a) the value of k,

k 的值；

(b) the values of x, y and z.

x、y 和 z 的值。

(a) ∵ kzyx

2

log

2

log

1

log 222

∴ log2x = k (1)

log2y = 2k (2)

log2z = 2k (3)

∵ 32z

xy

∴

(4) 5logloglog

32loglog

222

22

zyx

z

xy

By substituting (1), (2) and (3) into (4), we have

1

55

5)2(2

k

k

kkk

(b) ∵ 1log2 x

∴ 2x

∵ 2log2 y

∴ 4y

∵ 2log2 z

Question Bank

58

∴ 4

1z

3. (a) Given that the equation 0log

log

2log

loglog2

2

2

2

22 x

b

x

xa, where a 1 and b 1.

已知方程 0log

log

2log

loglog2

2

2

2

22 x

b

x

xa，其中 a 1 及 b 1。

(i) Set up a quadratic equation in x2log .

建立一個以 log2 x 為變數的二次方程。

(ii) If the equation has a double real root, express b in terms of a.

若該方程有一個二重實根，試以 a 表示 b。

(b) Hence, solve 0log

4

2log2

log2

22

2 xx

x.

由此，解 0log

4

2log2

log2

22

2 xx

x。

(a) (i)

bxbxa

x

b

x

xa

x

b

x

xa

x

b

x

xa

222

2

22

2

2

2

22

2

2

2

22

2

2

2

2

22

l o gl o g l o g)l o g(l o g 2

l o g 2

l o g

1l o g

l o g l o g

0l o g 2

l o g

1l o g

l o g l o g

0l o g

l o g

2l o g

l o g l o g

∴ 2 log2a (log2 x)2 – log2 b log2x + log2 b = 0

(1)

(ii) ∵ (1) has a double real root.

∴ = 0

∴ 8

8

82

8

22

8

222

22

2

2

1

0log

0loglog

0)loglog(log

0)log)(log 2(4)log(

ab

a

b

a

b

ab

abb

bab

(b)

0log

2log

2log

log 2log

0log

4

2log 2

log

2

2

4

2

2

22

2

22

2

xx

x

xx

x


59

4. Given that

9

11loga and

81

11logb .

已知

9

11loga 及

81

11logb 。

(a) Show that ba 13log43log22log3 .

證明 ba 13log43log22log3 。

(b) Express the following in terms of a and b.

試以 a 和 b 表示下列各題。

(i) log 3 (ii) log 2 (iii) log 72

(a) ∵

3 log 22 log 3

3 log2 log

9 log8 log

9

8log

9

11log

23

a

∴ 3 log 2 = 2 log 3 + a

∵

3 log 412 log 3

3 log10 log2 log

81 log10)(8 log

81

80 log

81

11 log

43

b

∴ 3 log 2 = 4 log 3 1 + b

∴ 3 log 2 = 2 log 3 + a = 4 log 3 1 + b

(b) (i) ∵ 3 log 2 = 2 log 3 + a = 4 log 3 1 + b

∴

3 log 21

13 log 43 log 2

ba

ba

∴ 2

13log

ba

(ii) By substituting log 3 = 2

1ba into

3 log 2 = 2 log 3 + a, we have

122 log 3

2

122 log 3

ba

aba

Question Bank

60

∴ 3

122log

ba

(iii)

223

)1()12(

2

12

3

123

3 log 22 log 3

)3(2 log72 log 23

ba

baba

baba

5. Solve the following simultaneous equations.

解下列各聯立方程。

(a)

4123

4343121 yx

yx

(b)

yx

yx

33log22)336(log

0016.05

(a)

(2) 4123

(1) 4343

121

yx

yx

From (1),

43433 1 yx (3)

From (2),

41423

41223

1

21

yx

yx

(4)

3 (4) (3),

2

44

164

8045

43413)433()423(3

2

11

y

y

y

y

yxyx


3

33

273

4343

3

2

x

x

x

x


(b)

(2) log 22)336(log

(1) 0016.05

33

yx

yx

From (1),


61

4

55

625

15

4

yx

yx

yx

From (2),

2

2

33

2

33

2

3

33

112

log)112(log

log2)112(log)3(log

log 22)112(3log

yx

yx

yx

yx


(4) 112

(3) 4

2

yx

yx

From (3),

x = y 4 (5)


0)1)(3(

032

32

11)4(2

2

2

2

yy

yy

yy

yy

y = 3 or y = 1 (rejected)


1

43

x


6. (a) Let n

n

nx

xT1

, where n is a positive integer. Prove that 211

nnn

TTTT for 3n .

設 n

nn

xxT

1 ，其中 n 是一個正整數。證明當 3n 時， 211 nnn TTTT 。

(b) Given that 41T . Hence, or otherwise, find the values of . and ,

432TTT

已知 41 T 。由此，或用其他方法，求 T2、T3 及 T4 的值。

(a)

L.H.S.

1

111

111

R.H.S

2

22

2

2

2

1

1

1

1

211

n

n

n

n

n

n

n

n

n

n

n

n

n

nn

T

xx

xx

xx

xx

xx

xx

xx

TTT

Question Bank

62

∴ 3for 211 nTTTT nnn

(b)

14

2)4(

2)(

12

1

1

2

21

2

2

22

T

xx

xx

xxT

52

4414

1123

TTTT

194

14452

2134

TTTT

7 Solve 819

119 4

x

xxx .

解 819

119 4

x

xxx 。

0]81)[( ]1)1[(

08)1(9])1[(

08)1(9)1(

08)1(9)1)(1(

8)1()1()1(9

8)1(9

)1()1(9

819

1419

4

3

4

3

4

3

24

3

4

3

2

3

4

3

2

1

2

1

2

1

4

3

2

1

4

1

2

1

xx

xx

xx

xxx

xxxx

xx

xx

xx

xx

or0

or11

or11

or1)1(

or1)1(

4

4

34

4

3

x

x

x

x

x

15

21

21

8)1(

8)1(

4

4

34

4

3

x

x

x

x

x


63

8. The figure shows the graphs of xy 2 , 2xy and kmxy . A(a, 8) lies on the graph of xy 2 .

The three graphs intersect at B and C respectively.

下圖所示為 y = 2x，y = x

2 及 y = mx + k 的圖像。A(a, 8) 位於 y = 2

x 的圖像上。該三個圖像分

別相交於 B 和 C。

(a) (i) Find the coordinates of A, B and C.

求 A、B 和 C 的坐標。

(ii) Find the values of m and k.

求 m 和 k 的值。

(b) Solve 0862 xx graphically.

利用圖解法解 0862 xx 。

(c) For 0 x 4.2, find the range of possible values of x that satisfies x2 2

x.

對於 0 x 4.2，求 x 值的可能範圍，使 x2 2

x。

(a) (i) ∵ The graph of y = 2x passes through A(a, 8).

∴

3

22

28

3

a

a

a

∴ The coordinates of A = 8) ,3(

Consider y = 2x.

When x = 2, y = 22 = 4

When x = 4, y = 24 = 16

Consider y = x2.

When x = 2, y = 22 = 4

When x = 4, y = 42 = 16

∴ The graphs of y = 2x and y = x2 intersect at (2, 4) and (4, 16).

∵ For 0 x 3, the graphs of y = 2x and

y = x2 intersect at B only.

∴ The coordinates of B = )4 ,2(

∵ For 3 < x 4.2, the graphs of y = 2x and

y = x2 intersect at C only.

Question Bank

64

∴ The coordinates of C = )16 ,4(

(ii) ∵ The graph of y = mx + k passes through B(2, 4) and C(4, 16).

∴ By substituting (2, 4) into y = mx + k, we have

4 = 2m + k (1)

By substituting (4, 16) into y = mx + k, we have

16 = 4m + k (2)

(2) (1), m212

∴ 6m

By substituting m = 6 into (1), we have

∴ 8

)6(24

k

k

(b)

86

086

2

2

xx

xx

From the graphs, the solution of 862 xx is 42 x .

∴ The solution of 0862 xx is 42 x .

(c) From the graphs, the required range is 42 x .


65

Multiple Choice Questions

多項選擇題

1. 23 )( nx

A. nx6

B. nx9

C. 2

6nx

D. 2

9nx

A

n

nn

x

xx

6

2323 )(

2 16x

A. 1

B. x

C. x2

D. x4

B

x

xx

4

2

116

16

3. 3 2

562

3

)(

a

aa

A. 3

5

a

B. 6

5

a

C. 3a

D. a6

C

3

3

2

6

5

2

3

3

2

6

5

2

3

3 2

562

3

)(

a

a

a

aa

a

aa

4.

ba

ba2

324 )(

A. 7

10

b

a

B. 5

10

b

a

C. 7

14

b

a

D. 5

14

b

a

C

7

14

714

16)2(12

2

612

2

3234

2

324 )(

b

a

ba

ba

ba

ba

ba

ba

ba

ba

5. Solve 04)4(316 xx .

解 04)4(316 xx 。

A. x = 1

B. x = 0 or 4

1

C. x = 0 or 1

D. x =4

1 or 1

A

Question Bank

66

0)14)(44(

04)4(3)4(

04)4(3)4(

04)4(316

2

2

xx

xx

xx

xx

4x = 4 or 4x = 1 (rejected)

1

44 1

x

x

6. If 3 21 2)8(4 xx , then x

若 3 21 2)8(4 xx ，則 x

A. 8 .

B. 8

5 .

C. 8

1.

D. 5.

C

18

239

3

213

22

2]2[2

2)8(4

3

2

13

3

2

)1(32

3 21

x

xx

xx

x

x

x

x

xx

∴ 8

1x

7. If 13363 xx , then x

若 13363 xx ，則 x

A. 1.

B. 2.

C. 3.

D. 4.

C

21

33

93

36)4(3

36)13(3

3633

3363

21

1

1

1

1

1

x

x

x

x

x

xx

xx

∴ 3x

8. Solve

)4(322

25

15

53

2

yx

yx

.

解聯立方程

)4(322

25

15

53

2

yx

yx

。

A. 0,2 yx

B. 2

1,1 yx

C. 2

3,1 yx

D. 2,2 yx

C

(2) )(4 322

(1) 25

15

53

2

yx

yx

From (1),

∴ 22

55 22

yx

yx

From (2),

∴ 42

123

22

)2(22

123

23

yx

yx

yx

yx


(4) 42

(3) 22

yx

yx

(3) + (4),

1

22

x

x


67


2

3

32

221

y

y

y

∴ The solution is x 1, 2

3y .

9. 64log22

A. 1

B. 2

C. 4

D. 8

C

4

)8(log64log 4

822

10. If 1log2log ab , then b =

若 1log2log ab ，則 b =

A. 10a2.

B. 10 + a2.

C. 10 + 2a.

D. 1 + 2a.

A

)(10 log log

log10 log log

log 21 log

1 log 2 log

2

2

ab

ab

ab

ab

∴ 210ab

11

aab

ba

log)log(

log2log

A. )log( ab

B. ab

C. 2

D. 2

1

C

)( log

)( log

)( log

)( log

log

)( log

log)( log

log log

log)( log

log 2 log

2

2

2

2

ba

ba

ba

ab

a

ab

ab

aab

ba

aab

ba

2

log

)( log 2

ba

ba

12. If a103 and b104 , then 6

5log

若 a103 及 b104 ，則 6

5log

A. ba 1 .

B. ba 1 .

C. ba 1 .

D. ba 1 .

C

3 = 10a and 4 = 10b

∴ log 3 = a and log 4 = b

ba

1

4 log3 log10 log

43

10 log

26

25 log

6

5 log

13. Solve )2(log2)1(log 22 xx .

解 )2(log2)1(log 22 xx 。

Question Bank

68

A. x = 2

B. x = 3

C. x = 2or 3

D. x = 3or 2

A

0)3)(2(

06

4)2)(1(

4log)]2)(1[(log

2)2(log)1(log

)2(log2)1(log

2

22

22

22

xx

xx

xx

xx

xx

xx

x = 2 or x = 3 (rejected)

14. If 129 x , then x =

若 129 x ，則 x =

A. 3log

4log.

B. 3

4log .

C. 9log

12log.

D. 12log9

1.

C

12 log9 log

12 log)9( log

129

x

x

x

∴ 9 log

12 logx

15. If kx 3log , then

2

1log

x

若 kx 3log ，則

2

1log

x

A. k6

B. 29k

C. k6

1

D. 29

1

k

A

∴ kx

kx

kx

kx

3 log

log 3

1

)( log

log

3

1

3

k

k

x

xx

6

)3(2

log 2

)( log1

log 2

2

16. Solve 02log)(log2 32 xx .

解 02log)(log2 32 xx 。

A. 100or 10

1x

B. 10or 100

1x

C. 100or 10x

D. 100or 10 x

A

0)2 log)(1 log 2(

02 log 3) (log2

02 log) 2(log

2

32

xx

xx

xx

100 or

10

1

2 logor 2

1 log

xx

xx

17. Solve

01log)2(log

)6(223

33

1

yx

yxx

.

解

01log)2(log

)6(223

33

1

yx

yxx

。

A. 2

5 ,

2

3 yx

B. 2

5 ,

2

3 yx


69

C. 2

3 ,

2

5 yx

D. 2

3 ,

2

5 yx

17. C

(2) 01log)2(log

(1) )6(223

33

1

yx

yxx

From (1),

yx

yxx

66

)6(22)23(

1

11

∴ yx 1 (3)

From (2),

yx

y

x

y

x

yx

63

3

12

3

1log

2log

1log)2(log

33

33

(4)

(4) (3),

052

0)1()63(

x

xx

2

5x

By substituting 2

5x into (3), we have

2

3

12

5

y

y

∴ The solution is 2

5x ,

2

3y .

18. Which of the following may represent the

graph of xy 2 ?

下列何者能代表 xy 2 的圖像？

A.

B.

C.

D.

A

Consider y = 2x.

When x = 0,

1

20

y

When x = 2,

1

4

22

y

Question Bank

70

19. Which of the following MUST not be a graph

of xy alog , where a > 0 and a 1?

下列何者不可能是 xy alog 的圖像？

（其中 a > 0 及 a 1。）

I.

II.

III.

A. II only 只有 II

B. III only 只有 III

C. I and II only 只有 I 及 II

D. II and III only 只有 II 及 III

B

Consider y = loga x.

∵

0

yax

∴ III must not be the required graph.

If a > 1, I may represent the required graph.

If 0 < a < 1, II may represent the required graph.

20. Which of the following functions is

represented by the graph shown below?

下列哪一個函數可由以下圖像所代表？

A. xy2

log

B. xy5.0

log

C. xy 2

D.

x

y

2

1

B

The graph passes through (1, 0).

∴ The graph may represent the graph of y = loga x,

where 0 < a < 1.


71

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