Determine if each statment is true or false. Justify your answer.

50. log 2 4 + log 2 8 = 5 log base 2 , 4 plus , log base 2 , 8 equals 5
51. log 3 3 2 = 1 2 log 3 3 log base 3 . 3 halves , equals , 1 half . log base 3 , 3
52. log ( x − 2 ) = log x log 2 log open x minus 2 close equals . fraction log x , over log 2 end fraction
53. log b x log b y = log b x y fraction log base b , x , over log base b , y end fraction . equals , log base b . x over y
54. ( log x ) 2 = log x 2 open log x close squared . equals log , x squared
55. log 4 7 − log 4 3 = log 4 4 log base 4 , 7 minus , log base 4 , 3 equals , log base 4 , 4

Write each logarithmic expression as a single logarithm.

56. 1 4 log 3 2 + 1 4 log 3 x 1 fourth . log base 3 , 2 plus , 1 fourth . log base 3 , x
57. 1 2 ( log x 4 + log x y ) − 3 log x z 1 half . open . log base x , 4 plus , log base x , y . close . minus 3 , log base x , z
58. x log 4 m + 1 y log 4 n − log 4 p x , log base 4 , m plus , 1 over y . log base 4 , n minus , log base 4 , p
59. ( 2 log b x 3 + 3 log b y 4 ) − 5 log b z open . fraction 2 , log base b , x , over 3 end fraction . plus . fraction 3 , log base b , y , over 4 end fraction . close . minus 5 , log base b , z

Expand each logarithm.

60. log 2 x y log . square root of fraction 2 x , over y end fraction end root
61. log s 7 t 2 log . fraction s square root of 7 , over t squared end fraction
62. log ( 2 x 5 ) 3 log . open , fraction 2 square root of x , over 5 end fraction , close cubed
63. log m 3 n 4 p − 2 log . fraction m cubed , over n to the fourth . p super negative 2 end super end fraction
64. log 4 4 r s 2 log 4 . square root of fraction 4 r , over s squared end fraction end root
65. log b x y 2 3 z 2 5 log base b . fraction square root of x , cube root of y squared end root , , over the fifth , root of z squared end root , end fraction
66. log 4 x 5 y 7 z w 4 log base 4 . fraction square root of x to the fifth , y to the seventh end root , over z , w to the fourth end fraction
67. log x 2 − 4 ( x + 3 ) 2 log . fraction square root of x squared , minus 4 end root , over open x plus 3 close squared end fraction

Write each logarithm as the quotient of two common logarithms. Do not simplify the quotient.

68. log 7 2 log base 7 , 2
69. log 3 8 log base 3 , 8
70. log 5 140 log base 5 , 140
71. log 9 3.3 log base 9 , 3.3
72. log 4 3 x log base 4 . 3 to the x

Astronomy The apparent brightness of stars is measured on a logarithmic scale called magnitude, in which lower numbers mean brighter stars. The relationship between the ratio of apparent brightness of two objects and the difference in their magnitudes is given by the formula m 2 − m 1 = − 2.5 log b 2 b 1 , m sub 2 , minus , m sub 1 , equals negative 2.5 log . fraction b sub 2 , over b sub 1 end fraction . comma  where m is the magnitude and b is the apparent brightness.

73. How many times brighter is a magnitude 1.0 star than a magnitude 2.0 star?
74. The star Rigel has a magnitude of 0.12. How many times brighter is Capella than Rigel?

C Challenge

Expand each logarithm.

75. log x 2 y 2 log . square root of fraction x square root of 2 , over y squared end fraction end root
76. log 3 [ ( x y 1 3 ) + z 2 ] 3 log base 3 . left bracket . open x , y super 1 third end super , close plus , z squared . right bracket cubed
77. log 7 r + 9 s 2 t 1 3 log base 7 . fraction square root of r plus 9 end root , over s squared . t super 1 third end super end fraction

Simplify each expression.

78. log 3 ( x + 1 ) − log 3 ( 3 x 2 − 3 x − 6 ) + log 3 ( x − 2 ) log base 3 , open x plus 1 close minus , log base 3 , open 3 , x squared , minus 3 x minus 6 close plus , log base 3 , open x minus 2 close
79. log ( a 2 − 10 a + 25 ) + 1 2 log 1 ( a − 5 ) 3 − log ( a − 5 ) log open , eh squared , minus 10 eh plus 25 close plus , 1 half log . fraction 1 , over open eh minus 5 close cubed end fraction . minus log open , square root of eh minus 5 end root , close

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