Math 2412 Activity 3(Due by EOC Mar. 23)

Graph the following exponential functions by modifying the graph of ( ) 2xf x = . Find the

range of each function.

1. ( ) 22xg x += 2. ( ) 2 2xg x = +

3. ( ) 22 1xg x += − 4. ( ) 2 xg x −=

5. ( ) 2xg x = − 6. ( ) 2 xg x −= −

Find a formula for the exponential function whose graph is given

7. 8.

Write each equation in its equivalent exponential form.

9. 26 log 64= 10. 92 log x= 11. 5log 125 y=

Write each equation in its equivalent logarithmic form.

12. 45 625= 13. 3 15

125

− = 14. 8 300y =

Simplify the following expressions.

15. 7log 49 16. 3log 27 17. 16 6

log

18. 13 9

log 19. 6log 6 20. 13 3

log

21. 81log 9 22. 6log 1 23. 7log 237

24. ( )5 2log log 32 25. ( )4 3 2log log log 8 26. 3

102 2

log 81 log 1

log 8 log .001

−

−

( )0,1

( )2,16

( )1,4

( )1, 6−

( )2, 36−

( )0, 1−

27. 1 12 3

log 16 log 9+ 28. 16 27log 4 log 3− 29. 1 1 17 8 22401 64 64

log log log+ +

30. 1 12 3

1 14 27

log log−

Graph the following logarithmic functions by modifying the graph of ( ) 2logf x x= . Find the

domain of each function.

31. ( ) ( )2log 2g x x= + 32. ( ) 22 logg x x= +

33. ( ) 22logg x x= − 34. ( ) ( )2logg x x= −

Expand each expression as much as possible, and simplify whenever possible.

35. ( )8log 64 7 36. ( )9log 9x 37. ( )10log 10,000x 38. 9

9log

x

39. 7logb x 40.

8

10log M − 41. 7

2log x 42. ( )3logb xy

43. 5log25

x

44. ( )

3 3

10 2

100 5log

3 7

x x

x

−

+

Compress each expression as much as possible, and simplify whenever possible.

45. 10 10log 250 log 4+ 46. 3 3log 405 log 5− 47. 2 2log 7logx y+

48. 10 107log 3logx y− 49. ( ) ( )4 4 4

1log log 2log 1

3x y x− + +

Use a calculator to evaluate the following logarithms to four places.

50. 6log 17 51. 16log 57.2 52. log 400

53. Simplify 5 2

10 10log 8 log 210

x x− , for 0x .

54. Find the value of x for 0a so that ( )( )2 3log log log 1a x =

Solve the following exponential equations.

55. 3 81x = 56. 2 13 27x+ = 57.

2 15

125

x− = 58. 2

67 7x−

=

59. 1 28 4x x− += 60.

24 3 4 2 0x x− + = 61. 23 5 3 24 0x x+ − =

Solve the following logarithmic equations.

62. 5log 3x = 63. ( )7log 2 2x + = − 64. 2log 4 2x + =

65. ( )6 6log 5 log 2x x+ + = 66. ( ) ( )2 2log 1 log 1 3x x− + + =

67. ( )2 22log 4 log 9 2x + = + 68. ( ) ( )10 10 10log 7 log 3 log 7 1x x+ − = +

69. ( ) ( )2 2 2

1log 1 log 3 logx x

x

− − + =

70. 1 1

3 3

log 8 log 27 log 81 2x + − = −

Solve the following mixed equations.

71. 2 45 5 125x x = 72. ( )5 5 5log 6 log 5 log 0x x− + − =

73. 2 12 23 9x x− = 74. 28 16x x+=

75. 1 13

9

x

x

+ = 76. 2 12 4x+ =

77. log 810

xx = 78.

3 3log logx x=

79. For what values of b is ( ) logbf x x= an increasing function? a decreasing function?

80. For , 0a b , , 1a b , find the value of ( ) ( )log loga bb a .

81. Find values of a and b so that the graph of ( ) bxf x ae= contains the points ( )3,10 and

( )6,50 .

82. Suppose that x is a number such that 3 4x = . Find the value of 23 x− .

83. Suppose that x is a number such that 1

23

x = . Find the value of 42 x− .

84. Suppose that x is a number such that 2 5x = . Find the value of 8x .

85. Suppose that x is a number such that 3 5x = . Find the value of 1

9

x

.

86. For , 0b y and 12

1,b , show that 2

loglog

1 log 2

bb

b

yy =

+.

Find a simplified formula for f g for the given functions.

87. ( ) ( ) 3

6log , 6 xf x x g x= =

88. ( ) ( ) 3 2

5log , 5 xf x x g x += =

89. ( ) ( )3

66 , logxf x g x x= =

90. ( ) ( )3 2

55 , logxf x g x 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 10x , you apply the function ( ) 12

f x x= to both sides of the

inequality. Since it’s an increasing function, the direction of the inequality is preserved, and

you get 5x .

b) To solve the inequality 2 10x− , you apply the function ( ) 12

f x x= − to both sides of the

inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and

you get 5x − .

c) To solve the inequality 13

1 2x , you apply the function ( ) 3f x x= to all sides of the

inequality. Since it’s an increasing function, the direction of the inequality is preserved, and

you get 3 6x .

d) To solve the inequality 13

1 2x − , you apply the function ( ) 3f x x= − to all sides of the

inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and

you get 3 6 6 3x x− − − − .

e) To solve the inequality 1 10 5x , you apply the function ( ) 10logf x x= to all sides of the

inequality. Since it’s an increasing function, the direction of the inequality is preserved, and

you get 10 10 10log 1 log 5 0 log 5x x .

f) To solve the inequality 1

1 52

x

, you apply the function ( ) 12

logf x x= to all sides of the

inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and

you get 1 1 12 2 2

log 1 log 5 log 5 0x x .

Use the previous discussion to solve the following inequalities(Use interval notation.):

91. 54 16− x 92. 2

2 1log 0

2

−

−

x

x 93.

2 4 55 5− x x 94. 1

12 1− x

95. 58 4 16x− 96. 12

2 11 log 0

2

x

x

− −

− 97. ( ) ( )

25 41 13 3

1x x−

98. 12

1log 1

1x

−

Simplify the following as much as possible:

99. 52 1

3 3 33log log log logA B A B− + +

100. ( )log

1log

ABC

ABC

101. 2

2

log 2log

1log log

A B

AAB

−

+

102. ( )log

10AB

103. ( )log log

100A B−

104. Find the exact value of the sum ( ) ( ) ( ) ( )999,99931 22 3 4 1,000,000

log log log log+ + + + .

105. Find the exact value of the sum ( ) ( ) ( ) ( )1,000,00032 41 2 3 999,999

log log log log+ + + + .

If all the terms of a sequence nx are positive, it is sometimes convenient to analyze the related

sequence 1n

n

x

x

+

. If 1 1n

n

x

x

+ , for n N , then the sequence nx is eventually nondecreasing.

If 1 1n

n

x

x

+ , for n N , then the sequence nx is eventually increasing. If 1 1n

n

x

x

+ , for n N ,

then the sequence nx is eventually nonincreasing. If 1 1n

n

x

x

+ , for n N , then the sequence

nx is eventually decreasing. Test the following sequences by analyzing 1n

n

x

x

+

.

106. 1

n

nx

n=

+ 107.

2 1n

nx

n

+= 108.

3

4

n

nx n

=

109. 2

2n n

nx = 110.

!

100n n

nx = 111.

( )( )

2

2 4 21

1 3 2 1n

nx

n n

=

−

112. ( )

( )

2 4 2

3 5 2 1n

nx

n

=

+ 113.

( )( )

2 4 2

1 3 2 1n

nx

n

=

− 114.

( )( )

2 4 2 1

1 3 2 1 2 2n

nx

n n

=

− +

115. Find an explicit formula for na in the following recursively defined sequence:

( )1 1

11,

1n na a a

n n+= = −

+.

{Hint: ( )

1 1 1

1 1n n n n= −

+ +, so

1

1 11

2 2a = = + , 2

1 1 1 11

2 3 2 3a

= − − = +

,

3

1 1 1 1 1 1

2 3 3 4 2 4a

= + − − = +

, 4

1 1 1 1 1 1

2 4 4 5 2 5a

= + − − = +

,…, see if you can

formulate the pattern.}

116. A sequence na is given recursively by 2

1 2n na a n+ = + . If 4 23a = , then what’s 1a ?

117. Find x so that 3,2 1,5 2x x x+ + + are consecutive terms of an arithmetic sequence.

118. Find x so that 2 ,3 2,5 3x x x+ + 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 n

a 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 n

a that generates the numbers in this arithmetic sequence.

c) What is the 99th number in this arithmetic sequence?

123. Find x so that , 2, 3x x x+ + are consecutive terms of a geometric sequence.

124. Find x so that 1, , 2x x x− + are consecutive terms of a geometric sequence.

125. a) Determine the common ratio for the following geometric sequence:

1 18,4,2,1, , ,

2 4

b) Find a formula for n

a that generates the numbers in this geometric sequence.

c) Is 1

1024 a number in this geometric sequence?

126. a) Determine the common ratio for the following geometric sequence:

1, 2,4, 8,16, 32,− − −

b) Find a formula for n

a 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. 4 8

23 9

+ + + 128. 2

6 23

+ + + 129. 3 9 27

14 16 64

− + − +

130. 64

9 12 163

+ + + + 131. ( )1

13

1

8k

k

−

=

132. ( )1

32

1

3k

k

−

=

133. ( )1

12

1

4k

k

−

=

− 134. ( )34

1

2k

k

=

Prove the following using the Principle of Mathematical Induction:

135. ( ) ( )1 5 9 4 3 2 1n n n+ + + + − = −

136. ( ) ( )3 5 7 2 1 2n n n+ + + + + = +

137. ( )( )3 1

1 4 7 3 22

n nn

−+ + + + − =

138. 2 1 3 11 3 3 3

2

n

n− −+ + + + =

139. 2 1 5 11 5 5 5

4

n

n− −+ + + + =

140. ( )( )

1 1 1 1

1 3 3 5 5 7 2 1 2 1 2 1

n

n n n+ + + + =

− + +

141. ( )

22

3 3 3 31

1 2 34

n nn

++ + + + =

142. ( )( )( )( )1 4 1

1 2 3 4 5 6 2 1 23

n n nn n

+ − + + + + − =

143. 3 2n n+ is divisible by 3

144. ( )( )1 2n n n+ + is divisible by 6

145. ( ) ( )( )1 12 2 2 2 2

11 2 3 4 1 1

2

n n n nn

− − +− + − + − = −

146. 1 1 1 1

1 1 1 1 11 2 3

nn

+ + + + = +

147. 2nn

148. 5n n− is divisible by 5

149. 7 4n n− is divisible by 3.

150. ( ) ( )

1 2 11

2! 3! 1 ! 1 !

n

n n+ + + = −

+ +

151. ( )2 3 11 2 2 3 2 4 2 2 1 2 1n nn n−+ + + + + = − +

152. 25 1n − is divisible by 8.

153. 9 4n n− is divisible by 5.

154. ( )( )2 !

2 6 10 14 4 2!

nn

n − =

155. 3 5n k j= + for some pair of natural numbers, k and j, for 8n .

156. ( )( )( ) ( )2

1 1 1 14 9 16

11 1 1 1 ; 2

2n

nn

n

+− − − − = .

157. 2 2 111 12n n+ ++ is divisible by 133, for 0n .

158.1 1 1 1 13

; 21 2 3 24

nn n n n n

+ + + + + + + +

.

159. ( )110 10 1n n n+ − + + is divisible by 81, for 0n .

Find a formula for the nth term of the following sequences:

160. 5 5 52 4 8, , , 161. 2 2 2

3 9 272, , , ,− − 162. 3,6, 12,24, 48,− − −

163. 1 1

4, .1n n

a a a+

= = 164. 121 1

2,n n

a a a+

= = − 165. 1 1

4,n n

a a a+

= = −

166. 1 1

11,

2n n

na a a

n+

+ = =

167.

1 1

2 51,

4

n

n

aa a

+

+ = =

168. In a geometric sequence of real number terms, the first term is 3 and the fourth term is 24.

Find the common ratio.

169. Find the seventh term of a geometric sequence whose third term is 94 and whose fifth term

is 8164 .

170. For what value(s) of k will 4, 1,2 2k k k− − − form a geometric sequence?

Find the sum of the following series:

171.

5

1

1

2 n

n

−

=

172. ( )5

113

1

3j

j

−

=

− 173. 1

2 2 2 23 9 27 3

2 n−+ + + + + +

174. ( )11 2 31n

e e e e− −− − −+ + + + + + 175. ( )1

4

2

n

n

=

176. ( )0

51

4

n

n

n

=

−

177.

0

5 1

2 3n n

n

=

+

178.1

0

2

5

n

n

n

+

=

179.

0

cos

5n

n

n

=

180.

0

1

2

n

n

=

181. 2

0

n

n

e

−

=

182.

0

2 1

3

n

n

n

=

−

183. Solve the equation

1

2 126

n

i

i=

= for n.

184. Find the second term of an arithmetic sequence whose first term is 2 and whose first, third,

and seventh terms form a geometric sequence.

185. 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.

186. The equation 1xx e= − 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

187. The equation 2x x 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

188. Consider the statement ( )( )2 1

1 2 32

n nn

+ −+ + + + = .

a) Show that if the statement is true for n k= , then it must be true for 1n k= + .

b) For which natural numbers is the statement true?

189. If ( )log 32

=b

b

b, then what’s the value of b?

190. The third term of an arithmetic sequence is 0. Find the sum of the first 5 terms.

191. The third term of a geometric sequence is 4. Find the product of the first 5 terms.

192. Is there a geometric sequence containing the terms 27, 8, and 12 in any order and not

necessarily consecutive? If so, give an example. If not, show why.

193. Is there a geometric sequence containing the terms 1, 2, and 5 in any order and not

necessarily consecutive? If so, give an example. If not, show why.

194. Find all numbers a and b so that 10, , ,a b ab are the first 4 terms of an arithmetic sequence.

195. Evaluate the sum 2 2 2 2 2 2 2 211 1 12 2 13 3 1000 990− + − + − + + − .

{Hint: ( )( )2 2a b a b a b− = − + .}

196. If 4 4 7x x−+ = , then what is the value of 8 8x x−+ ?

{Hint: 2 4 2 4 7 2 8 2 7 2x x x x x x x x− − + = + = and

2 4 2 4 7 2 2 8 7 2x x x x x x x x− − − − − − + = + = . So add the two equations together to

get ( ) ( )8 8 2 2 7 2 2 8 8 6 2 2x x x x x x x x x x− − − − −+ + + = + + = + . If you knew the value

of 2 2x x−+ , then you ‘d have the answer. Well, what’s ( )2

2 2x x−+ ?}

197. Assuming that 4log 6 A= , 4log 10 B= , and 4log 7 C= , then express the value of 4log 112

using A, B, or C.

{Hint: 2112 7 4= .}

198. Assuming that 4log 9 A= , 4log 5 B= , and 4log 6 C= , then express the value of 4

81log

5

using A, B, or C.

{Hint: 281 9= .}

199. Assuming that 5log 7 A= , 5log 12 B= , and 5log 9 C= , then express the value of 5log 245

using A, B, or C.

{Hint: 2245 5 7= .}

200. Assuming that 5log 9 A= , 5log 12 B= , and 5log 7 C= , then express the value of 5

9log

35

using A, B, or C.

{Hint: 35 5 7= .}

201. Let’s prove that log2 is an irrational number. Suppose that log2m

n= for positive integers

m and n. Then 10 2mn = and therefore 10 2m n= . Compare the ones digits of powers of 10

to the ones digits of powers of 2 to arrive at a contradiction.

202. Let’s prove that 5log 7 is an irrational number. Suppose that 5log 7

m

n= for positive

integers m and n. Then 5 7mn = and therefore 5 7m n= . Compare the ones digits of powers

of 5 to the ones digits of powers of 7 to arrive at a contradiction.

203. Let’s prove that 3log 5 is an irrational number. Suppose that 3log 5

m

n= for positive

integers m and n. Then 3 5mn = and therefore 3 5m n= . Compare the ones digits of powers

of 3 to the ones digits of powers of 5 to arrive at a contradiction.

204. a) Show that if , 0b d , then a c

b d is equivalent to ad bc .

{Hint: If a c

b d then 0 0

a c ad bc

b d bd

−− , and 0 0

ad bcad bc ad bc

bd

− − .}

b) Show that if , 0b d and a c

b d then

a a c c

b b d d

+

+.

c) Use Mathematical Induction to show that if 1 2, , , 0nb b b and 1 2

1 2

n

n

a a a

b b b , then

1 1 2

1 1 2

n n

n n

a a a a a

b b b b b

+ + +

+ + +.

Use the 1-1 functions f and g which are defined by the following tables to find the following.

x ( )f x x ( )g x

-1 1 -1 0

0 4 1 1

1 5 4 2

2 -1 10 -1

205. ( )( )4f g 206. ( )( )0g f 207. ( )( )1 1f g−

208. ( )( )1 1 2f g− − 209. ( )( )1 1 1g f− − − 210. Solve ( )1 10g x− = .

211. Solve ( )( ) 0f g x = . 212. Solve ( )( )1 1 0f g x− − = . 213. Solve( )( ) 2f g x .

214. Solve ( )( )1 1g f x− .

215. For the one-to-one functions ( ) ( ) ( ) 0,1 , 1,2 , 2,3f = and ( ) ( ) ( ) 1,0 , 2,1 , 4,3g = ,

( ) ( ) 1,1 , 2,2f g = and ( ) ( ) 0,0 , 1,1g f = , so they could be described as ( )( )f g x x=

and ( )( )g f x x= . Are f and g inverses? Explain.

216. You might think that if ( )( )f g x x= for all x in the domain of g and ( )( )g f x is defined

for all x in the domain of f , that ( )( )g f x x= . For ( ) ( ) 1,2 , 3,4g = and

( ) ( ) ( ) 2,1 , 4,3 , 6,3f = , ( )( )f g x x= for all x in the domain of g, ( )( )g f x is defined for

all x in the domain f . Is it the case that ( )( )g f x x= for all x in the domain of f? Explain.

217. A function is called an involution if it is its own inverse. Which of the following functions

are involutions? What type of symmetry does the graph of an involution have?

a) ( )f x x= b) ( )f x x= − c) ( ) ; 0f x x x= − d) ( )1

f xx

=

e) ( )2

1f x

x= f) ( )

2

2

1 ; 1 0

1 ;0 1

x xf x

x x

− − =

− −

g) ( )

1; 1 0

0 ; 0

1;0 1

x x

f x x

x x

+ −

= = −

218. Solve the equation 3 2log 4 log 8x x= .

219. What’s the sum of all the reciprocals of integers whose only prime factors are 2’s, 3’s, and

5’s?

1 1 1 1 1 1 1

1 2 3 5 2 3 2 5 3 5+ + + + + + +

{Hint: ( )( )( )1 1 1 1 1 1 1 1 1 12 3 5 2 3 5 2 3 2 5 3 5 2 3 5

1 1 1 1

+ + + = + + + + + + + .}

220. How many subsets of the set 2,3,4,5,6,7,8,9 contain at least one prime number?

221. What is the value of ( )100 100

1 1i j

i j= =

+

?

222. What’s the value of 111! 110!

109!

−?

223. The first three terms of a geometric sequence are 3 63, 3, 3 . What’s the fourth term?

224. For what value of x does 2 3 4

2 4 8 162log log log log log 40x x x x x+ + + + = ?

225. Suppose that 2 3a ar ar ar+ + + + and

2 3a as as as+ + + + are two different infinite

geometric series with the same first term. The sum of the first series is r, and the sum of

the second series is s. Find the numerical value of r s+ .

226. 1 1a = , 2

1

3a = , and 1

2

1

; 11

n nn

n n

a aa n

a a

++

+

+=

−. Find 2018a .

227. Three numbers are consecutive numbers of an arithmetic sequence with the first number

being 9. If 2 is added to the second number, and 20 is added to the third number, then the

three numbers will be consecutive numbers of a geometric sequence. What’s the smallest

possible value of the third number of the geometric sequence?

228. Find the domain of the function ( )( )( )2018 2017 2016 2015log log log log x .

229. If ( )3log 1xy = and ( )2log 1x y = , then find the value of ( )log xy .

230. In a geometric sequence with positive terms, each term is the sum of the next two terms.

What’s the common ratio?

231. Solve the equation 74 6

log log 16 log 8x xx − = − .

232. Solve the equation ( )( )( )10 10log log

10log 10,000x

x = .