Download Name: Math 2412 Activity 3(Due by Apr. 4) Graph the following

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. 53 
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 x1  27
59. 81 x  4x2
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 x2
2
75. 3x1 
77. x
log x
1
9x
76.
 108
78. log3 x  log3 x
22 x1  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 32 x .
1
83. Suppose that x is a number such that 2 x  . Find the value of 24 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   532 x
89. f  x   63 x , g  x   log 6 x
90. f  x   532 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
 x2 
91. 45 x  16
 2x  1 
96. 1  log 1 
0
2
 x2 
95. 8  45 x  16
2
1
5
94. 2 x1  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 Alog 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  n1  . If n1  1 , for n  N , then the sequence  xn  is eventually nondecreasing.
xn
 xn 
x
x
If n1  1 , for n  N , then the sequence  xn  is eventually increasing. If n1  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
xn1
 1 , for n  N , then the sequence
xn
x 
Test the following sequences by analyzing  n1  .
 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, an1  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 an1  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
n1
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.
 5n1 
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, an1   12 an
 n 1
155. a1  1, an1  
 an
 2n 
 2a  5 
156. a1  1, an1   n

 4 
151. 3,6, 12,24, 48,
154. a1  4, an1  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.
n2
166.

n 0

169.

n 0
n1
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
i1
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