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Name:_____________________ Math 2412 Activity 3(Due by Apr. 4) Graph the following exponential functions by modifying the graph of f x 2 x . Find the range of each function. 1. g x 2 x 2 2. g x 2 x 2 3. g x 2 x 2 1 4. g x 2 x 5. g x 2 x 6. g x 2 x Find a formula for the exponential function whose graph is given 7. 8. 2,16 0, 1 1, 6 1,4 0,1 2, 36 Write each equation in its equivalent exponential form. 9. 6 log 2 64 10. 2 log 9 x 11. log 5 125 y Write each equation in its equivalent logarithmic form. 12. 54 625 13. 53 1 125 14. 8 y 300 Simplify the following expressions. 15. log 7 49 16. log 3 27 17. log 6 16 18. log 3 19 19. log 6 6 20. log 3 1 3 21. log 81 9 23. 7log7 23 22. log 6 1 log3 81 log 1 log 2 2 8 log10 .001 24. log5 log 2 32 25. log 4 log3 log 2 8 26. 27. log 1 16 log 1 9 28. log16 4 log 27 3 1 log8 641 log 2 641 29. log 7 2401 2 3 30. log 1 14 log 1 2 3 1 27 Graph the following logarithmic functions by modifying the graph of f x log 2 x . Find the domain of each function. 31. g x log 2 x 2 32. g x 2 log 2 x 33. g x 2log 2 x 34. g x log 2 x Expand each expression as much as possible, and simplify whenever possible. 35. log8 64 7 36. log 9 9x 37. log10 10,000x 9 38. log 9 x 39. logb x7 40. log10 M 8 41. log 2 7 x 42. log b xy 3 x 43. log 5 25 100 x3 3 5 x 44. log10 2 3 x 7 Compress each expression as much as possible, and simplify whenever possible. 45. log10 250 log10 4 46. log 3 405 log 3 5 48. 7log10 x 3log10 y 49. 47. log 2 x 7log 2 y 1 log 4 x log 4 y 2log 4 x 1 3 Use a calculator to evaluate the following logarithms to four places. 50. log 6 17 51. log16 57.2 53. Simplify 10log10 8 x log10 2 x , for x 0 . 5 2 52. log 400 54. Find the value of x for a 0 so that log 2 log3 log a x 1 Solve the following exponential equations. 57. 52 x 55. 3x 81 56. 32 x1 27 59. 81 x 4x2 60. 42 x 3 4 x 2 0 1 125 58. 7 x 2 6 7 61. 32 x 5 3x 24 0 Solve the following logarithmic equations. 62. log 5 x 3 63. log7 x 2 2 65. log 6 x 5 log 6 x 2 64. log 2 x 4 2 66. log 2 x 1 log 2 x 1 3 67. 2log 2 x 4 log 2 9 2 68. log10 x 7 log10 3 log10 7 x 1 1 69. log 2 x 1 log 2 x 3 log 2 x 70. log x 8 log 1 27 log 1 81 2 3 3 Solve the following mixed equations. 71. 52 x 54 x 125 72. log5 6 log 5 x 5 log 5 x 0 73. 3x 12 92 x 74. 8x 16 x2 2 75. 3x1 77. x log x 1 9x 76. 108 78. log3 x log3 x 22 x1 4 79. For what values of b is f x logb x an increasing function? a decreasing function? 80. For a, b 0 , a, b 1, find the value of log a b logb a . 81. Find values of a and b so that the graph of f x aebx contains the points 3,10 and 6,50 . 82. Suppose that x is a number such that 3x 4 . Find the value of 32 x . 1 83. Suppose that x is a number such that 2 x . Find the value of 24 x . 3 84. Suppose that x is a number such that 2 x 5 . Find the value of 8 x . x 1 85. Suppose that x is a number such that 3 5 . Find the value of . 9 x 86. For b, y 0 and b 1, 12 , show that log 2b y logb y . 1 logb 2 Find a simplified formula for f g for the given functions. 87. f x log 6 x, g x 63 x 88. f x log 5 x, g x 532 x 89. f x 63 x , g x log 6 x 90. f x 532 x , g x log 5 x When solving inequalities, it is common to apply a function to all sides of the inequality. If the function is an increasing function, then the direction of the inequality is preserved, but if the function is a decreasing function, then the direction of the inequality is reversed. Here’s why: If f is increasing, then for x y , f x f y , and you see that the inequality direction is preserved. If f is decreasing, then for x y , f x f y , and you see that the inequality direction is reversed. Here are some examples: a) To solve the inequality 2 x 10 , you apply the function f x 12 x to both sides of the inequality. Since it’s an increasing function, the direction of the inequality is preserved, and you get x 5 . b) To solve the inequality 2 x 10 , you apply the function f x 12 x to both sides of the inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and you get x 5 . c) To solve the inequality 1 13 x 2 , you apply the function f x 3x to all sides of the inequality. Since it’s an increasing function, the direction of the inequality is preserved, and you get 3 x 6 . d) To solve the inequality 1 13 x 2 , you apply the function f x 3x to all sides of the inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and you get 3 x 6 6 x 3 . e) To solve the inequality 1 10 x 5 , you apply the function f x log10 x to all sides of the inequality. Since it’s an increasing function, the direction of the inequality is preserved, and you get log10 1 x log10 5 0 x log10 5 . x 1 f) To solve the inequality 1 5 , you apply the function f x log 1 x to all sides of the 2 2 inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and you get log 1 1 x log 1 5 log 1 5 x 0 . 2 2 2 Use the previous discussion to solve the following inequalities: 2x 1 92. log 2 0 x2 91. 45 x 16 2x 1 96. 1 log 1 0 2 x2 95. 8 45 x 16 2 1 5 94. 2 x1 1 2 97. 13 13 5 x2 4 x Simplify the following as much as possible: 99. log A3 log B 3 log A3 log B 3 1 93. 5x 4 x 55 1 1 98. log 1 1 2 x 1 100. 101. log ABC 1 log ABC log A2 2log B 1 log A2 log AB 102. 10 log AB 103. 100 log Alog B 104. Find the exact value of the sum log 12 log 32 log 43 999,999 log 1,000,000 . 105. Find the exact value of the sum log 12 log 23 log 34 log 1,000,000 999,999 . If all the terms of a sequence xn are positive, it is sometimes convenient to analyze the related x x sequence n1 . If n1 1 , for n N , then the sequence xn is eventually nondecreasing. xn xn x x If n1 1 , for n N , then the sequence xn is eventually increasing. If n1 1 , for n N , xn xn then the sequence xn is eventually nonincreasing. If xn is eventually decreasing. n 106. xn n 1 n2 109. xn n 2 112. xn 2 4 2n 3 5 2n 1 xn1 1 , for n N , then the sequence xn x Test the following sequences by analyzing n1 . xn n2 1 107. xn n n! 110. xn 100n 113. xn 2 4 2n 1 3 2n 1 3 108. xn n 4 n 1 2 4 2n 111. xn n 1 3 2n 1 114. xn 2 2 4 2n 1 1 3 2n 1 2n 2 115. Find an explicit formula for an in the following recursively defined sequence: a1 1, an1 an {Hint: 1 . n n 1 1 1 1 1 1 1 1 1 1 , so a1 1 , a2 1 , n n 1 n n 1 2 2 2 3 2 3 1 1 1 1 1 1 1 1 1 1 1 1 , a4 ,…, see if you can 2 3 3 4 2 4 2 4 4 5 2 5 formulate the pattern.} a3 116. A sequence an is given recursively by an1 2an n2 . If a4 23 , then what’s a1 ? 117. Find x so that x 3,2x 1,5x 2 are consecutive terms of an arithmetic sequence. 118. Find x so that 2x,3x 2,5x 3 are consecutive terms of an arithmetic sequence. 119. How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to get the sum 1092? 120. How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is -4 to get the sum 702? 121. a) Determine the common difference for the following arithmetic sequence: 5,1,7,13,19,25, b) Find a formula for an that generates the number in this sequence. c) Is 71 a number in this arithmetic sequence? 122. a) Determine the common difference for the following arithmetic sequence: 8,6,4,2,0, 2, b) Find a formula for an that generates the numbers in this arithmetic sequence. c) What is the 99th number in this arithmetic sequence? 123. Find x so that x, x 2, x 3 are consecutive terms of a geometric sequence. 124. Find x so that x 1, x, x 2 are consecutive terms of a geometric sequence. 125. a) Determine the common ratio for the following geometric sequence: 1 1 8,4,2,1, , , 2 4 b) Find a formula for an that generates the numbers in this geometric sequence. c) Is 1 a number in this geometric sequence? 1024 126. a) Determine the common ratio for the following geometric sequence: 1, 2,4, 8,16, 32, b) Find a formula for an that generates the numbers in this geometric sequence. c) Is 512 a number in this geometric sequence? Determine if the following geometric series converge or diverge. If it converges, find its sum. 127. 2 4 8 3 9 130. 9 12 16 133. 4 12 128. 6 2 64 3 131. 8 k 1 1 3 k 1 134. k 1 2 3 k 1 2 k 1 3 4 k 4n 3 n 2n 1 136. 3 5 7 2n 1 n n 2 137. 1 4 7 3n 2 138. 1 3 3 3 2 n1 n 3n 1 2 3n 1 2 3 9 27 4 16 64 132. 3 k 1 Prove the following using the Principle of Mathematical Induction: 135. 1 5 9 129. 1 3 2 k 1 139. 1 5 52 140. 5n1 1 1 1 1 3 3 5 5 7 141. 1 2 3 3 3 3 5n 1 4 1 2n 1 2n 1 n2 n 1 n 4 142. 1 2 3 4 5 6 n 2n 1 2 3 2n 1 2n n n 1 4n 1 3 143. n3 2n is divisible by 3 144. n n 1 n 2 is divisible by 6 145. 12 22 32 42 1 n 2 1 1 1 1 146. 1 1 1 1 2 3 n 1 n 1 n n 1 2 1 1 n 1 n 147. n 2n 148. n5 n is divisible by 5 Find a formula for the nth term of the following sequences: 149. 52 , 54 , 85 , 150. 2, 23 , 92 , 272 , 152. a1 4, an 1 .1an 153. a1 2, an1 12 an n 1 155. a1 1, an1 an 2n 2a 5 156. a1 1, an1 n 4 151. 3,6, 12,24, 48, 154. a1 4, an1 an 157. In a geometric sequence of real number terms, the first term is 3 and the fourth term is 24. Find the common ratio. 158. Find the seventh term of a geometric sequence whose third term is is 6481 . and whose fifth term 9 4 159. For what value(s) of k will k 4, k 1,2k 2 form a geometric sequence? Find the sum of the following series: 5 160. 5 1 n 161. 2 3 13 j 1 162. 2 23 92 272 163. 1 e e e 1 2 3 e n 1 164. 1 4 n 165. n2 166. n 0 169. n 0 n1 j 1 n 1 32 5 1 n n 2 3 1 2 167. n 0 2n 1 5n 168. n 0 170. e n 0 171. 5 4n cos n 5n 2 n n n 0 n 1 n 0 2n 1 3n 2 126 for n. n 172. Solve the equation i i1 173. Find the second term of an arithmetic sequence whose first term is 2 and whose first, third, and seventh terms form a geometric sequence. 174. The figure shows the first four of an infinite sequence of squares. The outermost square has an area of 4, and each of the other squares is obtained by joining the midpoints of the sides of the square before it. Find the sum of the areas of all the squares. 175. The equation x e x 1 has 0 as its only solution. If the Method of Successive Approximations is applied to approximate this solution, graphically indicate the result if the starting guess is a) a positive number b) a negative number 176. The equation x x2 x has 0 and 2 as its solutions. If the Method of Successive Approximations is applied to approximate these solutions, graphically indicate the result if the starting guess is a) greater than 2 b) between 0 and 2 c) less than 0 177. Consider the statement 1 2 3 n n 2 n 1 . 2 a) Show that if the statement is true for n k , then it must be true for n k 1. b) For which natural numbers is the statement true? 178. If logb 3b b , then what’s the value of b? 2