1 1 . . 0 0 n oo n i i t t c c e SS e s s n oo n i i t t u ......1 1 . . 0 0 n oo n i i t t c c e SS...

SSoolluuttiioonnss SSeeccttiioonn 00..11SSeeccttiioonn 00..11

1. 2(4 + (−1))(2 · − 4) = 2(3)(−8) = (6)(−8) = −48 2. 3 + ([4 − 2] · 9)

= 3 + (2 × 9) = 3 + 18 = 213. 20/(3*4) − 1 = 2012 − 1 = 53 − 1 = 23

4. 2 − (3*4)/10 = 2 − 1210 = 2 − 65 = 45

5. 3 + ([3 + (−5)])3 − 2 × 2

= 3 + (−2)3 − 4 = 1−1 = −16.

12 − (1 − 4)2(5 − 1) · 2 − 1 = 12 − (−3)16 − 1 = 1515 = 1

7. (2 − 5*(−1))/1 − 2*(−1) = 2 − 5 · (−1)1 − 2 · (−1) = 2 + 51 + 2 = 7 + 2 = 9

8. 2 − 5*(−1)/(1 − 2*(−1)) = 2 − 5 · (−1)1 − 2 · (−1) = 2 − −51 + 2 + 2 = 2 + 53 = 115

9. 2 · (−1)2/2 = 2 × (−1)22 = 2 × 12 = 22 = 1 10. 2 + 4 · 32 = 2 + 4 × 9 = 2 + 36 = 38

11. 2 · 42 + 1 = 2 × 16 + 1 = 32 + 1 = 33 12. 1 − 3 · (−2)2 × 2

= 1 − 3 × 4 × 2 = 1 − 24 = −2313. 3^2+2^2+1 = 32 + 22 + 1 = 9 + 4 + 1 = 14 14. 2^(2^2-2)

= 2(22−2) = 24−2 = 22 = 415.

3 − 2(−3)2−6(4 − 1)2

= 3 − 2 × 9−6(3)2 = 3 − 18−6 × 9 = −15−54 = 518

16. 1 − 2(1 − 4)22(5 − 1)2 · 2

= 1 − 2(−3)22(4)2 · 2 = 1 − 2 × 92 × 16 × 2

= 1 − 1864 = − 176417. 10*(1+1/10)^3

= 10(1 + 110)3 = 10(1.1)3 = 10 × 1.331 = 13.31

18. 121/(1+1/10)^2

= 121!1 + 110"2 = 1211.12

= 1211.21 = 100

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SSoolluuttiioonnss SSeeccttiioonn 00..11

19. 3[ −2 · 32−(4 − 1)2]

= 3[−2 × 9−32 ] = 3[−18−9 ] = 3 × 2 = 6

20. −[ 8(1 − 4)2−9(5 − 1)2]

= −[ 8 × 9−9 × 16] = −( 72−144) = −(− 12) = 12

21. 3[1 − (− 12)2]2 + 1 = 3[1 − 14]2 + 1 = 3[34]2 + 1 = 3[ 916] + 1 = 2716 + 1 = 4316

22. 3[19 − (23)2]2 + 1 = 3[19 − 49]2 + 1 = 3[−39 ]2 + 1 = 3[−13 ]2 + 1 = 3 19 + 1 = 39 + 1 = 43

23. (1/2)^2-1/2^2

= [12]2 − 122 = 14 − 14 = 024. 2/(1^2)-(2/1)^2

= 212 − [21]2 = 21 − 41 = −225. 3 × (2 − 5) = 3*(2-5) 26. 4 + 59 = 4+5/9 or 4+(5/9)

27. 32 − 5 = 3/(2-5)

Note 3/2-5 is wrong, as it corresponds to 32 − 5.28. 4 − 13 = (4-1)/3

29. 3 − 18 + 6 = (3-1)/(8+6)

Note 3-1/8-6 is wrong, as it corresponds to

3 − 18 − 6.30. 3 + 32 − 9 = 3+3/(2-9)

31. 3 − 4 + 78 = 3-(4+7)/8 32. 4 × 2!23"

= 4*2/(2/3) or (4*2)/(2/3)

33. 23 + + − +,2 = 2/(3+x)-x*y^2 34. 3 + 3 + ++, = 3+(3+x)/(x*y)

35. 3.1+3 − 4+−2 − 60+2 − 1 = 3.1x^3-4x^(-2)-60/(x^2-1)

36. 2.1+−3 − +−1 + +2 − 32 = 2.1x^(-3)-x^(-1)+(x^2-3)/2





37. !23"5

= (2/3)/5

38. 2!35"

= 2/(3/5)

39. 34−5 × 6 = 3^(4-5)*6Note that the entire exponent is in parentheses.

40. 23 + 57−9 = 2/(3+5^(7-9))Note that the entire exponent is in parentheses.

41. 3(1 + 4100)−3 = 3*(1+4/100)^(-3)

42. 3(1 + 4100 )−3 = 3((1+4)/100)^(-3)

43. 32+−1 + 4+ − 1 = 3^(2*x-1)+4^x-1

44. 2+2 − (22+)2 = 2^(x^2)-(2^(2*x))^2

45. 22+2−++1 = 2^(2x^2-x+1)Note that the entire exponent is in parentheses.

46. 22+2−+ + 1 = 2^(2x^2-x)+1Note that the entire exponent is in parentheses.

47. 4/−2+2 − 3/−2+

= 4*e^(-2*x)/(2-3e^(-2*x)) or 4(*e^(-2*x))/(2-3e^(-2*x)) or (4*e^(-2*x))/(2-3e^(-2*x))

48. /2+ + /−2+/2+ − /−2+

= (e^(2*x)+e^(-2*x))/(e^(2*x)-e^(-2*x))

49. 3(1 − (− 12)2)2 + 1 = 3(1-(-1/2)^2)^2+1

50. 3(19 − (23)2)2 + 1 = 3(1/9-(2/3)^2)^2+1





1. 33 = 27 2. (−2)3 = −83. −(2 · 3)2 = −(22 · 32) = −(4 · 9) = −36or−(2 · 3)2 = −62 = −36

4. (4 · 2)2 = 42 · 22 = 16 · 4 = 64 or

(4 · 2)2 = 82 = 64

5. (−23 )2 = (−2)232 = 49 6. (32)3 = 33

23 = 2787. (−2)−3 = 1(−2)3 = 1−8 = − 18 8. −2−3 = − 123 = − 189. (14)−2 = 1(1/4)2 = 11/16 = 16 10. (−23 )−2 = 1(−2/3)2 = 14/9 = 9411. 2 · 30 = 2 · 1 = 2 12. 3 · (−2)0 = 3 · 1 = 313. 2322 = 23+2 = 25 = 32or2322 = 8 · 4 = 32

14. 323 = 3231 = 32+1 = 33 = 27or

323 = 9 · 3 = 2715. 222−1242−4 = 22−1+4−4 = 21 = 2 16. 525−3525−2 = 52−3+2−2 = 5−1 = 1517. +3+2 = +3+2 = +5 18. +4+−1 = +4−1 = +3

19. −+2+−3, = −+2−3, = −+−1, = − ,+ 20. −+,−1+−1 = −+1−1,−1 = −+0,−1 = − 1,21. +3

+4 = +3−4 = +−1 = 1+ 22. ,5,3 = ,5−3 = ,2

23. +2,2+−1, = +2− (−1),2−1 = +3, 24. +−1,+2,2 = +−1−2,1−2 = +−3,−1 = 1+3,

25. (+,−103)2

+2,02 = +2(,−1)2(03)2+2,02

= +2,−206+2,02 = +2−2,−2−106−2

26. +2,02(+,0−1)−1 = +2,02

+−1,−1(0−1)−1 = +2,02

+−1,−10 = +2− (−1),1− (−1)02−1 = +3,204





= +0,−304 = 04,3

27. (+,−20+−10 )3 = (+,−20)3

(+−10)3 = +3,−603

+−303 = +3− (−3),−603−3 = +6,−600 = +6

,6

28. (+2,−100+,0 )

2 = (+2,−100)2(+,0)2

= +4,−2+2,202 = +4−2,−2−20−2

= +2,−40−2 = +2,402

29. (+−1,−202+, )

−2 = (+−1−1,−2−102)−2 = (+−2,−302)−2 = +4,60−4 = +4,6

04

30. ( +,−2+2,−10)

−3 = (+1−2,−2+10−1)−3 = (+−1,−10−1)−3 = +3,303

31. 3+−4 = 3+4 32. 12 +−4 = 12+4

33. 34 +−2/3 = 34+2/3 34. 45 ,−3/4 = 45,3/4

35. 1 − 0.3+−2 − 65 +−1 = 1 − 0.3+2 − 65+ 36. 13+−4 + 0.1+−23 = +43 + 0.13+2

37. 4√ = 2 38. 5√ ≈ 2.23639. 14√ = 1√4√ = 12 40. 19√ = 1√9√ = 1341. 169√ = 16√ 9√ = 43 42. 94√ = 9√4√ = 3243. 4√5 = 25 44. 625√ = 6545. 9√ + 16√ = 3 + 4 = 7 46. 25√ − 16√ = 5 − 4 = 147. 9 + 16√ = 25√ = 5 48. 25 − 16√ = 9√ = 3





49. 8 − 273√ = −193√ ≈ − 2.668 50. 81 − 164√ = 654√ ≈ 2.839

51. 27/83√ = 273√83√ = 32 52. 8 × 643√ = 83√ · 643√ = 2 · 4 = 8

53. (−2)2√ = 4√ = 2 54. (−1)2√ = 1√ = 1

55. 14 (1 + 15)√ = 164√ = 16√ 4√ = 42 = 2 56. 19 (3 + 33)√ = 369√ = 36√ 9√ = 63 = 2

57. 5262√ = 52√ 62√ = 56 58. 5262√ = 52√62√ = 56

59. (+ + 9)2√ = + + 9(+ + 9 > 0 because + is positive.)

60. ! + + 9√ "2 = + + 9

61. +3(53 + 63)3√ = +33√ (53 + 63)3√ = + (53 + 63)3√ (Not +(5 + 6))

62. +454644

√ = +44√544√ 644√ = +56

63. 4+,3+2,√ = 4,2+√ = 4√ ,2√+√ = 2,+√ 64. 4(+2 + ,2)82√ = 4√ +2 + ,2√

82√ = 2 +2 + ,2√ 8

(Not 2(+ + ,)/8)

65. 3√ = 31/2 66. 8√ = 81/2

67. +3√ = +3/268. +23√ = +2/3

69. +,23√ = (+,2)1/3 70. +2,√ = (+2,)1/2

71. +2+√ = +2

+1/2 = +2−1/2 = +3/2 72. ++√ = ++1/2 = +1−1/2 = +1/2

73. 35+2 = 35 +−2 74. 25+−3 = 25 +3





75. 3+−1.22 − 13+2.1 = 32 +−1.2 − 13 +−2.1 76. 23+−1.2 − +2.13 = 23 +1.2 − 13 +2.1

77. 2+3 − +0.12 + 43+1.1 = 23 + − 12 +0.1 + 43 +−1.1 78. 4+23 + +3/26 − 23+2 = 43 +2 + 16 +3/2 − 23 +−2

79. 3 +√4 − 53 +√ + 43+ +√ = 3+1/24 − 53+1/2 + 43+ · +1/2 = 34 +1/2 − 53 +−1/2 + 43 +−3/2

80. 35 +√ − 5 +√8 + 72 +3√

= 35+1/2 − 5+1/28 + 72+1/3 = 35 +−1/2 − 58 +1/2 + 72 +−1/3

81. 3 +25√4 − 72 +3√ = 3+2/54 − 72+3/2 = 34 +2/5 − 72 +−3/2

82. 18+ +√ − 23 +35√ = 18+3/2 − 23+3/5

= 18 +−3/2 − 23 +−3/5

83. 1(+2+1)3 − 34 +2 + 13√

= 1(+2+1)3 − 3(+2+1)1/3 = (+2+1)−3 − 34 (+2+1)−1/3

84. 23(+2+1)−3 − 3 (+2+1)73√ 4 = 23 (+2+1)3 − 34 (+2+1)7/3

85. 22/3 = 223√ 86. 34/5 = 345√

87. +4/3 = +43√ 88. ,7/4 = ,74√

89. (+1/2,1/3)1/5 = +√ ,3√5√ 90. +−1/3,3/2 = ,3/2

+1/3 = ,3√+3√

91. − 32 +−1/4 = − 32+1/4 = − 32 +4√ 92. 45 +3/2 = 4 +3√5

93. 0.2+−2/3 + 37+−1/2 = 0.2+2/3 + 3+1/27 = 0.2

+23√ + 3 +√794. 3.1+−4/3 − 117 +−1/7 = 3.1+4/3 − 117+1/7 = 3.1 +43√ − 11

7 +7√





95. 34(1 − +)5/2 = 34 (1 − +)5√ 96. 94(1 − +)−7/3 = 9(1 − +)7/3

4 = 9 (1 − +)73√ 497. 4−1/247/2 = 4−1/2+7/2 = 46/2 = 43 = 64 98. 21/5

22/5 = 21/5−2/5 = 2−1/5 = 121/5

99. 32/33−1/6 = 32/3−1/6 = 31/2 = 3√ 100. 21/32−122/32−1/3 = 21/3−1+2/3−1/3 = 2−1/3 = 121/3

101. +3/2+5/2 = +3/2−5/2 = +−1 = 1+ 102. ,5/4

,3/4 = ,5/4−3/4 = ,1/2 = ,√

103. +1/2,2+−1/2, = +1/2− (−1/2),2−1 = +, 104. +−1/2,+2,3/2 = +−1/2−2,1−3/2 = +−5/2,−1/2

105. (+,)1/3(,+)2/3 = +1/3,1/3 · ,2/3

+2/3 = +1/3−2/3,−1/3+2/3 = +−1/3,1/3 or (,+)1/3

106. (+,)−1/3(,+)1/3 = +−1/3,−1/3 · ,1/3

+1/3 +−1/3−1/3,1/3+1/3 = +−2/3,2/3 or (,+)2/3

107. +2 − 16 = 0 ⇒ +2 = 16 ⇒ + = ± 16√ ⇒ + = ± 4

108. +2 − 1 = 0 ⇒ +2 = 1 ⇒ + = ± 1√ ⇒ + = ± 1

109. +2 − 49 = 0 ⇒ +2 = 49 ⇒ + = ± 49√ ⇒ + = ± 23

110. +2 − 110 = 0 ⇒ +2 = 110 ⇒ + = ± 110√ ⇒ + = ± 110√

111. +2 − (1 + 2+)2 = 0 ⇒ +2 = (1 + 2+)2 ⇒ + = ± 1 + 2+If + = 1 + 2+, then −+ = 1 ⇒ + = −1.

If + = −(1 + 2+), then 3+ = −1 ⇒ + = − 13 .

So, + = 1 or − 13 .

112. +2 − (2 − 3+)2 = 0 ⇒ +2 = (2 − 3+)2 ⇒ + = ± 2 − 3+If + = 2 − 3+, then 4+ = 2 ⇒ + = 12 .

If + = −(2 − 3+), then −2+ = −2 ⇒ + = 1.

So, + = 1 or 12 .

113. +5 + 32 = 0 ⇒ +5 = −32 ⇒ + = −325√ = −2

114. +4 − 81 = 0 ⇒ +4 = 81 ⇒ + = ± 814√ = ± 3





115. +1/2 − 4 = 0 ⇒ +1/2 = 4 ⇒ + = 42 = 16

116. +1/3 − 2 = 0 ⇒ +1/3 = 2 ⇒ + = 23 = 8

117. 1 − 1+2 = 0 ⇒ 1 = 1+2 ⇒ +2 = 1 ⇒ + = ± 1√ = ± 1

118. 2+3 − 6+4 = 0 ⇒ 2+3 = 6+4 ⇒ 2+4 = 6+3 ⇒ 2+ = 6 ⇒ + = 3

119. (+ − 4)−1/3 = 2 ⇒ + − 4 = 2−3 = 18 ⇒ + = 4 + 18 = 338

120. (+ − 4)2/3 + 1 = 5 ⇒ (+ − 4)2/3 = 4 ⇒ + − 4 = ± 43/2 = ± 8 ⇒ + = 4 ± 8 = −4 or 12





19. (+2−2+ + 1)2 = (+2 − 2+ + 1)(+2 − 2+ + 1) = +2(+2 − 2+ + 1) − 2+(+2 − 2+ + 1) + 1(+2 − 2+ + 1) = +4 − 2+3 + +2 − 2+3 + 4+2 − 2+ + +2 − 2+ + 1 = +4 − 4+3 + 6+2 − 4+ + 120. (+ + , − +,)2 = (+ + , − +,)(+ + , − +,) = +(+ + , − +,) + ,(+ + , − +,) − +,(+ + , − +,) = +2 + +, − +2, + +, + ,2 − +,2 − +2, − +,2 + +2,2 = +2 + 2+, + ,2 − 2+2, − 2+,2 + +2,2

21. (,3 + 2,2 + ,)(,2 + 2, − 1) = ,3(,2 + 2, − 1) + 2,2(,2 + 2, − 1) + ,(,2 + 2, − 1) = ,5 + 2,4 − ,3 + 2,4 + 4,3 − 2,2 + ,3 + 2,2 − ,

1. +(4+ + 6) = 4+2 + 6+ 2. (4, − 2), = 4,2 − 2,3. (2+ − ,), = 2+, − ,2 4. +(3+ + ,) = 3+2 + +,5. (+ + 1)(+ − 3) = +2 + + − 3+ − 3 = +2 − 2+ − 3

6. (, + 3)(, + 4) = ,2 + 3, + 4, + 12 = ,2 + 7, + 12

7. (2, + 3)(, + 5) = 2,2 + 3, + 10, + 15 = 2,2 + 13, + 15

8. (2+ − 2)(3+ − 4) = 6+2 − 6+ − 8+ + 8 = 6+2 − 14+ + 8

9. (2+ − 3)2 = 4+2 − 12+ + 9 10. (3+ + 1)2 = 9+2 + 6+ + 111. (+ + 1+)2 = +2 + 2 + 1+2 12. (, − 1,)2 = ,2 − 2 + 1,2

13. (2+ − 3)(2+ + 3) = (2+)2 − 32 = 4+2 − 9 14. (4 + 2+)(4 − 2+) = 42 − (2+)2 = 16 − 4+2

15. (, − 1,)(, + 1,) = ,2 − (1,)2 = ,2 − 1,2 16. (+ − +2)(+ + +2) = +2 − (+2)2 = +2 − +4

17. (+2 + + − 1)(2+ + 4) = +2(2+ + 4) + +(2+ + 4) − 1(2+ + 4) = 2+3 + 4+2 + 2+2 + 4+ − 2+ − 4 = 2+3 + 6+2 + 2+ − 4

18. (3+ + 1)(2+2 − + + 1) = 3+(2+2 − + + 1) + 1(2+2 − + + 1) = 6+3 − 3+2 + 3+ + 2+2 − + + 1 = 6+3 − +2 + 2+ + 1





= ,5 + 4,4 + 4,3 − ,22. (+3 − 2+2 + 4)(3+2 − + + 2) = +3(3+2 − + + 2) − 2+2(3+2 − + + 2) + 4(3+2 − + + 2) = 3+5 − +4 + 2+3 − 6+4 + 2+3 − 4+2 + 12+2 − 4+ + 8 = 3+5 − 7+4 + 4+3 + 8+2 − 4+ + 823. (+ + 1)(+ + 2) + (+ + 1)(+ + 3) = (+ + 1)(+ + 2 + + + 3) = (+ + 1)(2+ + 5)

24. (+ + 1)(+ + 2)2 + (+ + 1)2(+ + 2) = (+ + 1)(+ + 2)(+ + 2 + + + 1) = (+ + 1)(+ + 2)(2+ + 3)

25. (+2+1)5(+ + 3)4 + (+2+1)6(+ + 3)3 = (+2+1)5(+ + 3)3(+ + 3 + +2 + 1) = (+2+1)5(+ + 3)3(+2 + + + 4)

26. 10+(+2+1)4(+3+1)5 + 15+2(+2+1)5(+3+1)4 = 5+(+2+1)4(+3+1)4[2(+3 + 1) + 3+(+2 + 1)] = 5+(+2+1)4(+3+1)4(5+3 + 3+ + 2)

27. (+3 + 1) + + 1√ − (+3+1)2 + + 1√ = (+3 + 1) + + 1√ · [1 − (+3 + 1)] = −+3(+3 + 1) + + 1√

28. (+2 + 1) + + 1√ − (+ + 1)3√ = + + 1√ · [+2 + 1 − (+ + 1)2√ ] = + + 1√ · [+2 + 1 − (+ + 1)] = (+2 − +) + + 1√ = +(+ − 1) + + 1√

29. (+ + 1)3√ + (+ + 1)5√ = (+ + 1)3√ · [1 + (+ + 1)2√ ] = (+ + 1)3√ · (1 + + + 1) = (+ + 2) (+ + 1)3√

30. (+2 + 1) (+ + 1)43√ − (+ + 1)73√ = (+ + 1)43√ · [+2 + 1 − (+ + 1)33√ ] = (+ + 1)43√ · [+2 + 1 − (+ + 1)] = (+2 − +) (+ + 1)43√ = +(+ − 1) (+ + 1)43√

31. a. 2+ + 3+2 = +(2 + 3+)b. +(2 + 3+) = 0 + = 0 or 2 + 3+ = 0 + = 0 or −2/3

32. a. ,2 − 4, = ,(, − 4)b. ,(, − 4) = 0 , = 0 or , − 4 = 0 , = 0 or 4





33. a. 6+3 − 2+2 = 2+2(3+ − 1)b. 2+2(3+ − 1) = 0 +2 = 0 or 3+ − 1 = 0 + = 0 or 1/3

34. a. 3,3 − 9,2 = 3,2(, − 3)b. 3,2(, − 3) = 0 ,2 = 0 or , − 3 = 0 , = 0 or 3

35. a. +2 − 8+ + 7 = (+ − 1)(+ − 7)b. (+ − 1)(+ − 7) = 0 + − 1 = 0 or + − 7 = 0 + = 1 or 7

36. a. ,2 + 6, + 8 = (, + 2)(, + 4)b. (, + 2)(, + 4) = 0 , + 2 = 0 or , + 4 = 0 , = −2 or −4

37. a. +2 + + − 12 = (+ − 3)(+ + 4)b. (+ − 3)(+ + 4) = 0 + − 3 = 0 or + + 4 = 0 + = 3 or −4

38. a. ,2 + , − 6 = (, − 2)(, + 3)b. (, − 2)(, + 3) = 0 , − 2 = 0 or , + 3 = 0 , = 2 or −3

39. a. 2+2 − 3+ − 2 = (2+ + 1)(+ − 2)b. (2+ + 1)(+ − 2) = 0 2+ + 1 = 0 or + − 2 = 0 + = −1/2 or 2

40. a. 3,2 − 8, − 3 = (3, + 1)(, − 3)b. (3, + 1)(, − 3) = 0 3, + 1 = 0 or , − 3 = 0 , = −1/3 or 3

41. a. 6+2 + 13+ + 6 = (2+ + 3)(3+ + 2)b. (2+ + 3)(3+ + 2) = 0 2+ + 3 = 0 or 3+ + 2 = 0 + = −3/2 or −2/3

42. a. 6,2 + 17, + 12 = (3, + 4)(2, + 3)b. (3, + 4)(2, + 3) = 0 3, + 4 = 0 or 2, + 3 = 0 , = −4/3 or −3/2

43. a. 12+2 + + − 6 = (3+ − 2)(4+ + 3)b. (3+ − 2)(4+ + 3) = 0 3+ − 2 = 0 or 4+ + 3 = 0 + = 2/3 or −3/4

44. a. 20,2 + 7, − 3 = (4, − 1)(5, + 3)b. (4, − 1)(5, + 3) = 0 4, − 1 = 0 or 5, + 3 = 0 , = 1/4 or −3/5

45. a. +2 + 4+, + 4,2 = (+ + 2,)2b. (+ + 2,)2 = 0 + + 2, = 0 + = −2,

46. a. 4,2 − 4+, + +2 = (2, − +)2b. (2, − +)2 = 0 2, − + = 0 , = +/2





47. a. +4 − 5+2 + 4 = (+2 − 1)(+2 − 4) = (+ + 1)(+ − 1)(+ + 2)(+ − 2)b. (+ + 1)(+ − 1)(+ + 2)(+ − 2) = 0 + + 1 = 0 or + − 1 = 0 or

+ + 2 = 0 or + − 2 = 0 + = ± 1 or ±2

48. a. ,4 + 2,2 − 3 = (,2 − 1)(,2 + 3) = (, + 1)(, − 1)(,2 + 3)b. (, + 1)(, − 1)(,2 + 3) = 0 , + 1 = 0 or , − 1 = 0 or ,2 + 3 = 0 , = ± 1 (Notice that ,2 + 3 = 0 has no real solutions.)





1. + − 4+ + 1 · 2+ + 1+ − 1 = (+ − 4)(2+ + 1)(+ + 1)(+ − 1) = 2+2 − 7+ − 4+2 − 1

2. 2+ − 3+ − 2 · + + 3+ + 1 = (2+ − 3)(+ + 3)(+ − 2)(+ + 1) = 2+2 + 3+ − 9+2 − + − 2

3. + − 4+ + 1 + 2+ + 1+ − 1 = (+ − 4)(+ − 1) + (+ + 1)(2+ + 1)(+ + 1)(+ − 1) = 3+2 − 2+ + 5+2 − 1

4. 2+ − 3+ − 2 + + + 3+ + 1 = (2+ − 3)(+ + 1) + (+ − 2)(+ + 3)(+ − 2)(+ + 1) = 3+2 − 9+2 − + − 2

5. +2+ + 1 − + − 1+ + 1 = +2 − (+ − 1)+ + 1 = +2 − + + 1+ + 1

6. +2 − 1+ − 2 − 1+ − 1 = (+2 − 1)(+ − 1) − (+ − 2)(+ − 2)(+ − 1) = +3 − +2 − 2+ + 3+2 − 3+ + 2

7. 1! ++−1"

+ + − 1 = + − 1+ + + − 1 = + − 1 + +(+ − 1)+ = +2 − 1+

8. 2(+−2+2 ) − 1+ − 2 = 2+2+ − 2 − 1+ − 2

= 2+2 − 1+ − 29. 1+[+ − 3+, + 1,] = 1+[+ − 3 + ++, ] = 2+ − 3+2,

10. ,2+ [2+ − 3, + +,] = ,2

+ [2+ − 3 + ++, ] = ,2(3+ − 3)+, = ,(3+ − 3)+ = 3+, − 3,+

11. (+ + 1)2(+ + 2)3 − (+ + 1)3(+ + 2)2

(+ + 2)6

= (+ + 1)2(+ + 2)2[(+ + 2) − (+ + 1)](+ + 2)6 = (+ + 1)2

(+ + 2)4

12. 6+(+2+1)2(+3+2)3 − 9+2(+2+1)3(+3+2)2

(+3+2)6 = 3+(+2+1)2(+3+2)2[2(+3 + 2) − 3+(+2 + 1)]

(+3+2)6 = 3+(+2+1)2(−+3 − 3+ + 4)

(+3+2)4

13. (+2 − 1) +2 + 1√ − +4

+2+1√+2 + 1 = (+2 − 1)(+2 + 1) − +4

(+2 + 1) +2 + 1√ = −1(+2+1)3√

14. + +3 − 1√ − 3+4

+3−1√+3 − 1 = +(+3 − 1) − 3+4(+3 − 1) +3 − 1√

= −2+4 − +(+3−1)3√ = −+(2+3 + 1)

(+3−1)3√





15.

1(++,)2 − 1+2, = +2 − (+ + ,)2,+2(+ + ,)2

= +2 − +2 − 2+, − ,2,+2(+ + ,)2 = −,(2+ + ,)

,+2(+ + ,)2 = −(2+ + ,)+2(+ + ,)2

16.

1(++,)3 − 1+3, = +2 − (+ + ,)2,+2(+ + ,)2

= +3 − +3 − 3+2, − 3+,2 − ,3,+3(+ + ,)3

= −,(3+2 + 3+, + ,2),+3(+ + ,)3 = −(3+2 + 3+, + ,2)

+3(+ + ,)3





1. + + 1 = 0 ⇒ + = 0 − 1 ⇒ + = −1 2. + − 3 = 1 ⇒ + = 1 + 3 ⇒ + = 43. −+ + 5 = 0 ⇒ − + = −5 ⇒ + = 5 4. 2+ + 4 = 1 ⇒ 2+ = −3 ⇒ + = − 325. 4+ − 5 = 8 ⇒ 4+ = 13 ⇒ + = 134 6. 34 + + 1 = 0 ⇒ 34 + = −1 ⇒ + = − 437. 7+ + 55 = 98 ⇒ 7+ = 43 ⇒ + = − 437 8. 3+ + 1 = + ⇒ 2+ = −1 ⇒ + = − 129. + + 1 = 2+ + 2 ⇒ − + = 1 ⇒ + = −1 10. + + 1 = 3+ + 1 ⇒ − 2+ = 0 ⇒ + = 011. 5+ + 6 = 8 ⇒ 5+ = 8 − 6 ⇒ + = 8 − 65 12. + − 1 = 8+ + < ⇒ (1 − 8)+ = < + 1 ⇒ + = < + 11 − 813. 2+2 + 7+ − 4 = 0 ⇒ (2+ − 1)(+ + 4) = 0 ⇒ + = −4, 12

14. +2 + + + 1 = 0 ⇒ Δ = 62 − 458 = −3 < 0,so this equation has no real solutions.

15. +2 − + + 1 = 0 ⇒ Δ = 62 − 458 = −3 < 0,so this equation has no real solutions.

16. 2+2 − 4+ + 3 ⇒ Δ = 62 − 458 = −8 < 0,so this equation has no real solutions.

17. 2+2 − 5 = 0 ⇒ +2 = 52 ⇒ + = ± 52√ 18. 3+2 − 1 = 0 ⇒ +2 = 13 ⇒ + = ± 13√

19. −+2 − 2+ − 1 = 0 ⇒ − (+ + 1)2 = 0 ⇒ + = −1

20. 2+2 − + − 3 = 0 ⇒ (2+ − 3)(+ + 1) = 0 ⇒ + = 32, −1

21. 12 +2 − + − 32 = 0 ⇒ +2 − 2+ − 3 = 0 ⇒ (+ + 1)(+ − 3) = 0 ⇒ + = −1, 3

22. − 12 +2 − 12 + + 1 = 0 ⇒ +2 + + − 2 = 0 ⇒ (+ + 2)(+ − 1) = 0 ⇒ + = −2, 1

23. +2 − + = 1 ⇒ +2 − + − 1 = 0 ⇒ + = −6 ± 62 − 458√25 = 1 ± 5√2

24. 16+2 = −24+ − 9 ⇒ 16+2 + 24+ + 9 = 0 (4+ + 3)2 = 0 ⇒ + = − 34





25. + = 2 − 1+ ⇒ +2 = 2+ − 1 ⇒ +2 − 2+ + 1 = 0 ⇒ (+ − 1)2 = 0 ⇒ + = 1

26. + + 4 = 1+ − 2 ⇒ (+ + 4)(+ − 2) = 1 ⇒ +2 + 2+ − 8 = 1 ⇒ +2 + 2+ − 9 − 0 ⇒ + = −6 ± 62 − 458√25 = −2 ± 40√2 = −1 ± 10√

27. +4 − 10+2 + 9 = 0 ⇒ (+2 − 1)(+2 − 9) = 0 ⇒ +2 = 1 or +2 = 0 ⇒ + = ± 1, ± 3 28. +4 − 2+2 + 1 = 0 ⇒ (+2−1)2 = 0

⇒ + = ± 129. +4 + +2 − 1 = 0 ⇒ +2 = −6 ± 62 − 458√25 ⇒ +2 = −1 ± 5√2 ⇒ + = ± −1 ± 5√2√

30. +3 + 2+2 + + = 0 ⇒ +(+2 + 2+ + 1) = 0 ⇒ +(+ + 1)2 = 0 ⇒ + = 0, −1

31. +3 + 16+2 + 11+ + 6 = 0 ⇒ (+ + 1)(+ + 2)(+ + 3) = 0 ⇒ + = −1, −2, −3 32. +3 − 6+2 + 12+ − 8 = 0

⇒ (+ − 2)3 = 0 ⇒ + = 233. +3 + 4+2 + 4+ + 3 = 0 ⇒ (+ + 3)(+2 + + + 1) = 0 ⇒ + = −3(For +2 + + + 1 = 0, Δ = 62 − 458 = −3 < 0, sothere are no real solutions to this quadratic equation.)

34. ,3 + 64 = 0 ⇒ ,3 = −64 ⇒ , = −643√ = −4

35. +3 − 1 = 0 ⇒ +3 = 1 ⇒ + = 13√ = 1

36. +3 − 27 = 0 ⇒ +3 = 27 ⇒ + = 273√ = 3

37. ,3 + 3,2 + 3, + 2 = 0 ⇒ (, + 2)(,2 + , + 1) = 0 ⇒ , = −2(For ,2 + , + 1 = 0, Δ = 62 − 458 = −3 < 0, sothere are no real solutions to this quadratic equation.)

38. ,3 − 2,2 − 2, − 3 = 0 ⇒ (, − 3)(,2 + , + 1) = 0 ⇒ , = 3(For ,2 + , + 1 = 0, Δ = 62 − 458 = −3 < 0, so thereare no real solutions to this quadratic equation.)

39. +3 − +2 − 5+ + 5 = 0 ⇒ (+ − 1)(+2 − 5) = 0 ⇒ + = 1, ± 5√

40. +3 − +2 − 3+ + 3 = 0 ⇒ (+ − 1)(+2 − 3) = 0 ⇒ + = 1, ± 3√





41. 2+6 − +4 − 2+2 + 1 = 0 ⇒ (2+2 − 1)(+4 − 1) = 0 ⇒ (2+2 − 1)(+2 − 1)(+2 + 1) = 0(Think of the cubic you get by substituting , for +2)

⇒ + = ± 1, ± 12√

42. 3+6 − +4 − 12+2 + 4 = 0 ⇒ (3+2 − 1)(+4 − 4) = 0 ⇒ (3+2 − 1)(+2 − 2)(+2 + 2) = 0(Think of the cubic you get by substituting , for +2)

⇒ + = ± 2√ , ± 13√

43. (+2 + 3+ + 2)(+2 − 5+ + 6) = 0 ⇒ (+ + 2)(+ + 1)(+ − 2)(+ − 3) = 0 ⇒ + = −2, −1, 2, 3

44. (+2−4+ + 4)2(+2+6+ + 5)3 = 0 ⇒ (+ − 2)4(+ + 1)3(+ + 5)3 = 0 ⇒ + = −5, −1, 2





1. +4 − 3+3 = 0, +3(+ − 3) = 0, + = 0, 3 2. +6 − 9+4 = 0, +4(+2 − 9) = 0, + = 0, ±33. +4 − 4+2 = −4, +4 − 4+2 + 4 = 0,

(+2−2)2 = 0, + = ± 2√4. +4 − +2 = 6, +4 − +2 − 6 = 0,

(+2 − 3)(+2 + 2) = 0, + = ± 3√

5. (+ + 1)(+ + 2) + (+ + 1)(+ + 3) = 0, (+ + 1)(+ + 2 + + + 3) = 0,

(+ + 1)(2+ + 5) = 0, + = −1, −5/26. (+ + 1)(+ + 2)2 + (+ + 1)2(+ + 2) = 0, (+ + 1)(+ + 2)(+ + 2 + + + 1) = 0, (+ + 1)(+ + 2)(2+ + 3) = 0, + = −1, −2, −3/2

7. (+2+1)5(+ + 3)4 + (+2+1)6(+ + 3)3 = 0, (+2+1)5(+ + 3)3(+ + 3 + +2 + 1) = 0, (+2+1)5(+ + 3)3(+2 + + + 4) = 0, + = −3(Neither +2 + 1 = 0 nor +2 + + + 4 = 0 has a realsolution.)

8. 10+(+2+1)4(+3+1)5 − 10+2(+2+1)5(+3+1)4 = 0, 10+(+2+1)4(+3+1)4[+3 + 1 − +(+2 + 1)] = 0, 10+(+2+1)4(+3+1)4(1 − +) = 0, + = −1, 0, 1

9. (+3 + 1) + + 1√ − (+3+1)2 + + 1√ = 0, (+3 + 1) + + 1√ [1 − (+3 + 1)] = 0, −+3(+3 + 1) + + 1√ = 0, + = 0, −1

10. (+2 + 1) + + 1√ − (+ + 1)3√ = 0, + + 1√ [+2 + 1 − (+ + 1)] = 0, (+2 − +) + + 1√ = 0, +(+ − 1) + + 1√ = 0, + = −1, 0, 1

11. (+ + 1)3√ + (+ + 1)5√ = 0, (+ + 1)3√ (1 + + + 1) = 0, (+ + 2) (+ + 1)3√ = 0, + = −1(+ = −2 is not a solution because (+ + 1)3√ is not

defined for + = −2.)

12. (+2 + 1) (+ + 1)43√ − (+ + 1)73√ = 0, (+ + 1)43√ [+2 + 1 − (+ + 1)] = 0, (+2 − +) (+ + 1)43√ = 0, +(+ − 1) (+ + 1)43√ = 0, + = −1, 0, 1

13. (+ + 1)2(2+ + 3) − (+ + 1)(2+ + 3)2 = 0, (+ + 1)(2+ + 3)(+ + 1 − 2+ − 3) = 0, (+ + 1)(2+ + 3)(−+ − 2) = 0, + = −2, −3/2, −1

14. (+2−1)2(+ + 2)3 − (+2−1)3(+ + 2)2 = 0, (+2−1)2(+ + 2)2(+ + 2 − +2 + 1) = 0, −(+2−1)2(+ + 2)2(+2 − + − 3) = 0, + = −2, −1, 1, (1 ± 13√ )/2





15. (+ + 1)2(+ + 2)3 − (+ + 1)3(+ + 2)2

(+ + 2)6 = 0,

(+ + 1)2(+ + 2)2[(+ + 2) − (+ + 1)](+ + 2)6 = 0,

(+ + 1)2(+ + 2)4 = 0, (+ + 1)2 = 0,

+ = −1

16. 6+(+2+1)2(+2+2)4 − 8+(+2+1)3(+2+2)3

(+2+2)8 = 0,

2+(+2+1)2(+2+2)3[3(+2 + 2) − 4(+2 + 1)](+2+2)8 = 0,

−2+(+2+1)2(+2 − 2)

(+2+2)5 = 0, −2+(+2+1)2(+2 − 2) = 0, + = 0, ± 2√

17. 2(+2 − 1) +2 + 1√ − +4

+2+1√+2 + 1 = 0,

2(+2 − 1)(+2 + 1) − +4(+2 + 1) +2 + 1√ = 0,

+4 − 2(+2 + 1) +2 + 1√ = 0, +4 − 2 = 0, + = ± 24√

18. 4+ +3 − 1√ − 3+4

+3−1√+3 − 1 = 0,

4+(+3 − 1) − 3+4(+3 − 1) +3 − 1√ = 0,

+4 − 4+(+3 − 1) +3 − 1√ = 0, +4 − 4+ = 0, +(+3 − 4) = 0, + = 0, ± 43√

19. + − 1+ = 0, +2 − 1 = 0, + = ± 1 20. 1 − 4+2 = 0, +2 − 4 = 0, + = ± 2

21. 1+ − 9+3 = 0, +2 − 9 = 0, + = ± 3 22. 1+2 − 1+ + 1 = 0, + + 1 − +2 = 0, +2 − + − 1 = 0, + = (1 ± 5√ )/2

23. + − 4+ + 1 − ++ − 1 = 0,

(+ − 4)(+ − 1) − +(+ + 1)(+ + 1)(+ − 1) = 0, −6+ + 4(+ + 1)(+ − 1) = 0, −6+ + 4 = 0, + = 2/3

24. 2+ − 3+ − 1 − 2+ + 3+ + 1 = 0,

(2+ − 3)(+ + 1) − (2+ + 3)(+ − 1)(+ − 1)(+ + 1) = 0, −2+(+ − 1)(+ + 1) = 0, −2+ = 0, + = 0

25. + + 4+ + 1 + + + 43+ = 0,

3+(+ + 4) + (+ + 1)(+ + 4)3+(+ + 1) = 0,

(+ + 4)(3+ + + + 1)3+(+ + 1) = 0,

(+ + 4)(4+ + 1)3+(+ + 1) = 0,

26. 2+ − 3+ − 2+ − 3+ + 1 = 0,

(2+ − 3)(+ + 1) − +(2+ − 3)+(+ + 1) = 0,

(2+ − 3)(+ + 1 − +)+(+ + 1) = 0,

(2+ − 3)+(+ + 1) = 0, 2+ − 3 = 0, + = 3/2





(+ + 4)(4+ + 1) = 0, + = −4, −1/4





1. ? (0, 2), @(4, −2), A(−2, 3), B(−3.5, −1.5), C (−2.5, 0), D(2, 2.5)2. ? (−2, 2), @(3.5, 2), A(0, −3), B(−3.5, −1.5), C (2.5, 0), D(−2, 2.5)

5. Solve the equation + + , = 1 for , to get , = 1 − +. Then plot some points:

Graph:

3. 4.

+ −2 −1 0 1 2, = 1− + 3 2 1 0 −1





6. Solve the equation , − + = −1 for , to get , = −1 + +. Then plot some points:

Graph:

7. Solve the equation 2, − +2 = 1 for , to get , = 1 + +22 . Then plot some points:

Graph:

+ −2 −1 0 1 2, = −1 + + −3 −2 −1 0 1

+ −2 −1 0 1 2, = 1 + +2

2 2.5 1 0.5 1 2.5





8. Solve the equation 2, + +√ = 1 for , to get , = 1 − +√2 . Then plot some points:

Graph:

9. Solve the equation +, = 4 for , to get , = 4+ . Then plot some points:

Graph:

+ 0 1 4 9 16, = 1− +√2 0.5 0 −0.5 −1 −1.5

+ −3 −2 −1 1 2 3, = 4+ − 43 −2 −4 4 2 43





10. Solve the equation +2, = −1 for , to get , = − 1+2 . Then plot some points:

Graph:

11. Solve the equation +, = +2 + 1 for , to get , = + + 1+ . Then plot some points:

Graph:

+ −3 −2 −1 1 2 3, = − 1+2 − 19 − 14 −1 −1 − 14 − 19

+ −2 −1 − 12 12 1 2, = + + 1+ − 52 −2 − 52 52 2 52





12. Solve the equation +, = 2+3 + 1 for , to get , = 2+2 + 1+ . Then plot some points:

Graph:

+ −2 −1 − 12 12 1 2, = 2+2 + 1+ 152 1 − 32 52 3 172

13. (2 − 1)2 + (−2 + 1)2√ = 2√ 14. (6 − 1)2 + (1 − 0)2√ = 26√

15. (0 − 5)2 + (6 − 0)2√ = 52 + 62√ 16. (6 − 5)2 + (6 − 0)2√ = 2(6 − 5)2√ or 2√ |6 − 5|17. Set the two distances equal and solve:

(1 − 0)*2 + (E − 0)2√ = (1 − 2)2 + (E − 1)2√ ⇒ 1 + E2√ = 2 − 2E + E2√ ⇒ 1 + E2 = 2 − 2E + E2 ⇒ 2E = 1 ⇒ E = 12

18. Set the two distances equal and solve:

(E + 1)*2 + (E − 0)2√ = (E − 0)2 + (E − 2)2√ ⇒ 2E2 + 2E + 1√ = 2E2 − 4E + 4√ ⇒ 2E2 + 2E + 1 = 2E2 − 4E + 4 ⇒ 6E = 3 ⇒ E = 12

19. Circle with center (0, 0) and radius 3. 20. The single point (0, 0).





1.

2.

3.

4.

5. log416 = the power to which we need to raise 4 in order to get 16. Because 4 2 = 16, this power is 2, so log416 = 2.6. log5125 = the power to which we need to raise 5 in order to get 125. Because 5 3 = 125, this power is 3, so

log5125 = 3.

7. log5 125 = the power to which we need to raise 5 in order to get 125 . Because 5 −2 = 125 , this power is −2, so

log5 125 = −2.

ExponentialForm 102 = 100 43 = 64 44 = 256 0.450 = 1 81/2 = 2 2√ 4−3 = 164

LogarithmicForm log10100 = 2 log464 = 3 log4256 = 4 log0.451 = 0 log82 2√ = 12 log4( 164) = −3

ExponentialForm 101 = 10 55 = 3,125 63 = 216

LogarithmicForm log1010 = 1 log53,125 = 5 log6216 = 3

ExponentialForm 0.53 = 0.125 41/4 = 2√ 3−4 = 181

LogarithmicForm log0.50.125 = 3 log4 2√ = 14 log3( 181) = −4

ExponentialForm 0.32 = 0.09 (12)0 = 1 10−3 = 0.001

LogarithmicForm log0.30.09 = 2 log1/21 = 0 log100.001 = −3

ExponentialForm 9−2 = 181 210 = 1,024 64−1/3 = 14

LogarithmicForm log9 181 = −2 log21,024 = 10 log64 14 = − 13

ExponentialForm 2−1 = 12 105 = 100,000 10−5 = 0.00001

LogarithmicForm log2 12 = −1 log10100,000 = 5 log100.00001 = −5

ExponentialForm 211 = 2,048 16−1/4 = 12 0.252 = 0.0625

LogarithmicForm log22,048 = 11 log16 12 = − 14 log0.250.0625 = 2






log4 164 = −3.

9. log 100,000 = log10100,000 = the power to which we need to raise 10 in order to get 100,000. Because

10 5 = 100,000, this power is 5, so log 100,000 = 5.

10. log 1,000 = log101,000 = the power to which we need to raise 10 in order to get 1,000. Because 10 3 = 1,000, this

power is 3, so log 1,000 = 3.

11. log1616 = the power to which we need to raise 16 in order to get 16. Because 16 1 = 16, this power is 1, so

log1616 = 1.

12. log1/2 12 = the power to which we need to raise 12 in order to get 12 . Because (12) 1 = 12, this power is 1, so

log1/2 12 = 1.


log4 116 = −2.


log2 18 = −3.

15. log2 2√ = the power to which we need to raise 2 in order to get 2√ . Because 2 1/2 = 2√ , this power is 12, so

log2 2√ = 12 .

16. log4 2√ = the power to which we need to raise 4 in order to get 2√ . Because 4 1/4 = 2√ , this power is 14, so

log4 2√ = 14 .

17. By Identity (1), log63 + log64 = log6(3 × 4) = log6 12.18. By Identity (2), log63 − log64 = log6 34 .19. log62 − log65 − log64 = log62 − (log65 + log64) = log62 − (log6(5 × 4)) (Identity 1)





= log6( 25 × 4) (Identity 2) = log6 11020. log63 + log62 − log67 = log6(3 × 2) − log67 (Identity 1) = log6(3 × 27 ) (Identity 2) = log6 6721. log63 − 3log62 = log63 − log623 (Identity 3) = log6( 323) (Identity 2) = log6 3822. 3log62 + 2log63 = log623 + log632 (Identity 3) = log6(23 × 32) (Identity 1) = log6 7223. 4log6 + + 5log6 , = log6 +4 + log6 ,5 (Identity 3) = log6 +4,5 (Identity 1)

24. 3log6 G − 2log6 H = log6 G3 − log6 H2 (Identity 3) = log6 G3H2 (Identity 2)

25. 2log6 + + 3log6 , − 4log6 0 = log6 +2 + log6 ,3 − log6 04 (Identity 3)

= log6(+2,3) − log6 04 (Identity 1) = log6 +2,304 (Identity 2)

26. 4log6 G − 3log6 H − 2log6 I = log6 G4 − log6 H3 − log6 I2 (Identity 3)

= log6 G4 − (log6 H3 + log6 I2) = log6 G4 − log6(H3I2) (Identity 1) = log6 G4H3I2 (Identity 2)

27. + log62 − 2log6 + = log62+ − log6 +2 (Identity 3) = log6 2++2 (Identity 2)

28. G log6 H + H log6 G = log6 HG + log6 GH (Identity 3) = log6 HGGH (Identity 1)

29. log 21 = log(3 × 7) = log 3 + log 7 = 6 + 8 (Identity 1)

30. log 14 = log(2 × 7) = log 2 + log 7 = 5 + 8 (Identity 1)

31. log 42 = log(2 × 3 × 7) = log 2 + log 3 + log 7 = 5 + 6 + 8 (Identity 1)

32. log 28 = log(2 × 2 × 7) = log 2 + log 2 + log 7 = 25 + 8 (Identity 1)

33. log(17) = − log 7 = −8 (Identity 5)





34. log(13) = − log 3 = −6 (Identity 5)

35. log(23) = log 2 − log 3 = 5 − 6 (Identity 2)

36. log(72) = log 7 − log 2 = 8 − 5 (Identity 2)

37. log(47) = log 4 − log 7 (Identity 2) = log 2 + log 2 − log 7 (Identity 1) = 25 − 8

38. log(29) = log 2 − log 9 (Identity 2) = log 2 − (log 3 + log 3) (Identity 1) = 5 − 2639. log 16 = log24 = 4log 2 = 45 (Identity 3)

40. log 81 = log34 = 4log 3 = 46 (Identity 3)

41. log 0.03 = log( 3102) = log 3 − log 102 = log 3 − 2log 10 = 6 − 242. log 7,000 = log(7 × 103) = log 7 + log 103 = log 7 + 3log 10 = 8 + 343. log 5 = log(102 ) = log 10 − log 2 = 1 − 5 (Identity 2)

44. log 25 = log(1004 ) = log 100 − log 4 = log 102 − log22 = 2log 10 − 2log 2 = 2 − 25

45. log 7√ = log(71/2) = 12 log 7 = 8246. log( 23√ ) = log(2) − log 3√ = log(2) − log(31/2) = log 2 − 12 log 3 = 5 − 6247. 4 = 2+ is exactly the equation we solve in order to calculate log24; answer: 2. alternatively, write the equation

4 = 2+ in logarithmic form to get + = log24 = 2.48. 81 = 3+ is exactly the equation we solve in order to calculate log381; answer: 4. alternatively, write the equation

81 = 3+ in logarithmic form to get + = log381 = 4.





49. Take the base 3 logarithm of both sides of the given equation 27 = 32+−1 to get

log327 = log332+−13 = (2+ − 1)log33 By Identity 3

3 = 2+ − 1 By Identity 4

+ = 3 + 12 = 2 Solve for +.

50. Take the base 4 logarithm of both sides of the given equation 42−3+ = 256 to get

log442−3+ = log4256(2 − 3+)log44 = 4 By Identity 3

2 − 3+ = 4 By Identity 4

+ = 2 − 43 = − 23 Solve for +.

51. Take the base 5 logarithm of both sides of the given equation 5−++1 = 1125 to get

log55−++1 = log5( 1125)(−+ + 1)log55 = − log5125 By Identities 3 and 5

−+ + 1 = −3 By Identity 4

+ = 1 + 3 = 4 Solve for +.

52. Take the base 3 logarithm of both sides of the given equation 3+2−12 = 127 to get

log33+2−12 = log3( 127)(+2 − 12)log33 = − log327 By Identities 3 and 5

+2 − 12 = −3 By Identity 4

+ = ± 9√ = ±3 Solve for +.

53. First divide both sides of the given equation by 50 to get 2.4 = 23J. Take the common logarithm of both sides:

log 2.4 = log(23J)log 2.4 = 3J log 2 By Identity 3

3J = log 2.4log 2 Divide.

J = log 2.43log 2 ≈ 0.4210 Solve for J.If instead you take the base-2 logarithm, the answer is represented as

log2(2.4)3 .54. First divide both sides of the given equation by 500 to get 2 = 1.12J. Take the common logarithm of both sides:





log 2 = log(1.12J)log 2 = 2J log 1.1 By Identity 3

2J = log 2log 1.1 Divide.

J = log 22log 1.1 ≈ 3.6363 Solve for J.

55. First divide both sides of the given equation by 300 to get 103 = 1.34J−1. Take the common logarithm of both

sides: log(103 ) = log(1.34J−1)log 10 − log 3 = (4J − 1)log 1.3 By Identities 2 and 3

4J − 1 = log 10 − log 3log 1.3 Divide.

J = 14(log 10 − log 3log 1.3 + 1) ≈ 1.3972 Solve for J.

56. First divide both sides of the given equation by 700 to get 1007 = 1.043J+1. Take the common logarithm of both

sides: log(1007 ) = log(1.043J+1)log 100 − log 7 = (3J + 1)log 1.04 By Identities 2 and 3

3J + 1 = log 100 − log 7log 1.04 Divide.

J = 13(log 100 − log 7log 1.04 − 1) ≈ 22.2675 Solve for J.




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