1. Application of
Logarithms

2. Click on the title below you wish to view
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I. Compound Interest Formula
II. Continuous Compound Interest Formula
III. Population Growth Formula
IV. Doubling-Time Growth Formula
V. Exponential Decay
VI. Half-Life Decay Formula
VII. Newton’s Law of Cooling

3. Compound
Interest Formula

5. P is the Principle invested
The amount of money invested in a Savings Institution
The value of an object:
House
New Car
Piece of Property
Bicycle

6. total value of the investment after t years
A is the
total value of the investment after t years

7. number of years the Principle has been invested
t is the
number of years the Principle has been invested

8. r is the
Rate of interest
Expressed as a decimal

9. Number of times the interest is
n is the
Number of times the interest is
Compounded per year
“Compounded annually”
n = 1

10. Number of times the interest is
n is the
Number of times the interest is
Compounded per year
“Compounded semi-annually”
n = 2

11. Number of times the interest is
n is the
Number of times the interest is
Compounded per year
“Compounded quarterly”
n = 4

12. Number of times the interest is
n is the
Number of times the interest is
Compounded per year
“Compounded monthly”
n = 12

13. Example

14. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

15. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

16. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

17. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

18. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

19. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

20. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

21. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

22. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

23. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

24. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.

25. Substitute

27. Calculator Ready Form

29. One thousand dollars is invested at 12% interest compounded annually.
Determine how much the investment is worth
after 2 years.
The investment is worth
$1,254.40

30. Next
Example

31. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.

32. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.
You Determine

33. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.

34. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.

35. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.

36. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.

37. The value of a new $500 television
Decreases 10% per year.
Find its value after
5 years.

38. Substitute

40. Calculator Ready Form

41. The television is worth $295.25

42. Next
Example

43. One hundred dollars is invested at 7.2% interest compounded quarterly.
How long will it take for the investment to double?

44. You Determine One hundred dollars is invested at 7.2% interest
compounded quarterly.
How long will it take for the investment to double?
You Determine

45. One hundred dollars is invested at 7.2% interest compounded quarterly.
How long will it take for the investment to double?

46. One hundred dollars is invested at 7.2% interest compounded quarterly.
How long will it take for the investment to double?

47. One hundred dollars is invested at 7.2% interest compounded quarterly.
How long will it take for the investment to double?

48. One hundred dollars is invested at 7.2% interest compounded quarterly.
How long will it take for the investment to double?

49. t = One hundred dollars is invested at 7.2% interest
compounded quarterly.
How long will it take for the investment to double?
unknown t =

50. t = One hundred dollars is invested at 7.2% interest
compounded quarterly.
How long will it take for the investment to double?
unknown t =

51. Substitute t =
unknown

52. t =
unknown

55. Exponential Expression
Isolate
I s o l a t e
Exponential Expression

56. LOG IT

57. Calculator Ready Form

58. It takes approximately 10 years
Nearest Year
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It takes approximately 10 years

59. Continuous Compound
Interest Formula

60. Where did this formula come from?

61. If Compounded Continually
Then how many times per year?
Then n is approaching infinity

62. n approaches infinity r remains constant x approaches infinity Then
Reciprocal
n approaches infinity
r remains constant
Then
x approaches infinity

65. Reciprocal of Both Sides

67. Study

68. x approaches infinity

69. x approaches infinity
2nd TABLE

70. x approaches infinity x 4 8 12
100
365
10,000
100,000
1,000,000
2.4414
2.5658
2.6130
2.7048
2.7146
2.7182
2.7183

71. x approaches infinity x 4 8 12
100
365
10,000
100,000
1,000,000
2.4414

72. x approaches infinity 2.4414 2.5658 x 4 8 12 100 365 10,000 100,000
1,000,000
2.4414
2.5658

73. x approaches infinity 2.4414 2.5658 2.6130 x 4 8 12 100 365 10,000
100,000
1,000,000
2.4414
2.5658
2.6130

74. x approaches infinity 2.4414 2.5658 2.6130 2.7048 x 4 8 12 100 365
10,000
100,000
1,000,000
2.4414
2.5658
2.6130
2.7048

75. x approaches infinity 2.4414 2.5658 2.6130 2.7048 2.7146 x 4 8 12 100
365
10,000
100,000
1,000,000
2.4414
2.5658
2.6130
2.7048
2.7146

76. x approaches infinity 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 x 4 812
100
365
10,000
100,000
1,000,000
2.4414
2.5658
2.6130
2.7048
2.7146
2.7182

77. x approaches infinity x 4 8 12
100
365
10,000
100,000
1,000,000
2.4414
2.5658
2.6130
2.7048
2.7146
2.7182
2.7183

78. x approaches infinity approaching 2.4414 2.5658 2.6130 2.7048 2.714612
100
365
10,000
100,000
1,000,000
2.4414
2.5658
2.6130
2.7048
2.7146
2.7182
2.7183
approaching

79. x approaches infinity

80. x approaches infinity

81. Continuous Compound
Interest Formula

82. Next
Example

83. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

84. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

85. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

86. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

87. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

88. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

89. Mrs. Johnson received a bonus equivalent to 10% of her yearly salary
and has decided to deposit it in a savings account
in which interest is compounded continuously.
Her salary is $38,500 per year and
the account pays 7.5% interest.
How long will it take for her deposit to double in value?

90. Substitute

91. Exponential Expression
Isolate
I s o l a t e
Exponential Expression

93. 1
To the nearest year
Calculator Ready Form

94. 1
To the nearest year

95. Next
Example

96. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for?

97. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for?

98. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for? ?

99. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for? ?

100. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for? ?

101. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for? ?

102. Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday.
What annual interest rate compounded continually should the parent look for?

103. Substitute

104. Exponential Expression
Isolate
I s o l a t e
Exponential Expression

106. 1

107. Nearest hundredth of a percent
Calculator Ready Form

108. =
8.05%

109. Next
Example

110. How long will it take to double and investment at 4.5% interest
Doubling Time
How long will it take to double and investment at 4.5% interest

111. How long will it take to double and investment at 4.5% interest

112. How long will it take to double and investment at 4.5% interest

113. How long will it take to double and investment at 4.5% interest

114. How long will it take to double and investment at 4.5% interest

115. How long will it take to double and investment at 4.5% interest

116. How long will it take to double and investment at 4.5% interest

117. How long will it take to double and investment at 4.5% interest
Substitute

118. How long will it take to double and investment at 4.5% interest
Solve for t to the nearest hundredth

120. 1

122. How long will it take to double and investment at 4.5% interest
Solve for t to the nearest hundredth

123. Next
Example

124. You solve for r to the nearest tenth of a percent
What interest rate should an investor seek to double his money in 20 years?
You solve for r to the nearest tenth of a percent
You Try Using

125. You solve for r to the nearest tenth of a percent
What interest rate should an investor seek to double his money in 20 years?
You solve for r to the nearest tenth of a percent

126. You Try

127. Will invests $2000 in a bond trust that pays 9% interest compounded semiannually.
His friend Henry invests $2000 in a Certificate of Deposit that pays 8 ½ % compounded continuously.
Who will have more money after 20 years, Will or Henry? How much more money?

128. Will invests $2000 in a bond trust that pays 9% interest compounded semiannually.
His friend Henry invests $2000 in a Certificate of Deposit that pays 8 ½ % compounded continuously.
Who will have more money after 20 years, Will or Henry? How much more money?

129. Will
$684.84

130. You Try

131. Pat, who is 35, also invests $2000 in an IRA.
At the age of 25 Coris invests $ in an Individual Retirement Account (IRA) that is allowed to accumulate interest tax-free until she retires.
Pat, who is 35, also invests $2000 in an IRA.
Pat and Coris each earn 8% annually compounded continuously, and each withdraws the funds from the account at age 65.
To the nearest hundred, how much more money does Coris collect than Pat?

132. At the age of 25 Coris invests $ in an Individual Retirement Account (IRA) that is allowed to accumulate interest tax-free until she retires.
Pat, who is 35, also invests $2000 in an IRA.
Pat and Coris each earn 8% annually compounded continuously, and each withdraws the funds from the account at age 65.
To the nearest hundred, how much more money does Coris collect than Pat?
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$27,000

133. Population
Growth

134. is the size of the original population

135. P is the size of the population
after t years

136. r annual growth rate
of the population

137. t is the number of years

138. Example

139. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987
a) Using these statistics, find the average growth rate of the population in Raleigh-Durham (to the nearest tenth of a percent) and determine an equation of the population growth curve for this region.
b) Assuming a constant growth rate, predict the population for Raleigh-Durham in 1995 to the nearest unit.

140. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987
a) Using these statistics, find the average growth rate of the population in Raleigh-Durham (to the nearest tenth of a percent) and determine an equation of the population growth curve for this region.
b) Assuming a constant growth rate, predict the population for Raleigh-Durham in 1995 to the nearest unit.
Find r

141. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

142. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

143. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

144. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

145. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987
unknown
560,774
665,400

146. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987
unknown
560,774 7
665,400

147. Find the population growth rate for Raleigh-Durham, North Carolina to the nearest tenth of a percent
unknown
560,774 7
665,400

148. unknown
560,774 7
665,400

152. OR
Law 2

154. Nearest tenth of a percent
Calculator Ready Form
Nearest tenth of a percent

155. Next Step

156. We need to find an equation that MODELS the population growth for Raleigh-Durham, North Carolina
You Find

157. This is the equation that MODELS the population growth in Raleigh-Durham, North Carolina

158. Next Step

159. Use this equation to project the population in Raleigh-Durham, North Carolina in to the nearest unit

160. t To find P we need to know the value of ?
Use this equation to project the population in Raleigh-Durham, North Carolina in to the nearest unit
To find P we need to know
the value of ? t

161. Use this equation to project the population in Raleigh-Durham, North Carolina in to the nearest unit
The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987
t = 15

162. Use this equation to project the population in Raleigh-Durham, North Carolina in to the nearest unit

163. Link Back to Table of Contents
Use this equation to project the population in Raleigh-Durham, North Carolina in to the nearest unit
Link Back to Table of Contents

164. Doubling-Time
Growth
Formula

165. is the size of the
Original Population
Bacteria
Culture of Yeast
ETC.

166. Population after t time
is the size of the
Population after t time
Years
Weeks
Minutes
Months
Hours
Seconds
ETC.

167. is the amount of time it takes for the population to double
Years
Weeks
Minutes
Months
Hours
Seconds
ETC.

168. is the time period
Years
Weeks
Minutes
Months
Hours
Seconds
ETC.

169. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

170. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

171. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

172. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

173. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

174. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

175. A certain bacteria population doubles in size every 12 hours
A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

176. Substitute

180. The population grows by a factor of 16 in 2 days
Link Back to Table of Contents
The population is 16 times greater than the size of the original population
The population grows by a factor of in 2 days

181. EXPONENTIAL
DECAY

182. Graphs

183. Exponential Growth
Exponential Decay

184. Exponential Growth
Exponential Decay

185. Exponential Growth
Exponential Decay

186. Exponential Growth
Exponential Decay

187. Exponential Growth
Exponential Decay

188. General Form Exponential Equation
a is a constant
Exponential Growth
Exponential Decay

189. You Try

190. You may want check your answers with your graphing calculator
Determine whether each equation represents an exponential growth or an exponential decay curve.
You may want check your answers with your graphing calculator

191. Exponential Growth Exponential Decay Exponential Growth
Determine whether each equation represents an exponential growth or an exponential decay curve.
You may want check your answers with your graphing calculator
Exponential Growth
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Exponential Decay
Exponential Growth
Exponential Growth

192. Half-Life
Decay
Formula

193. Radioactive Substances
Used for
Radioactive Substances

194. Rewrite

196. is the original quantity of the radioactive substance
(isotope)

197. A is the amount of radioactive substance
after t years
Smaller
Larger or Smaller Amount?

198. of the radioactive substance
h is the half-life
of the radioactive substance

199. t is the number of years

200. The half life of radioactive radium (226Ra) is 1599 years.
Given an initial quantity of grams, how much will remain after 1000 years.

201. The half life of radioactive radium (226Ra) is 1599 years
The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of grams, how much will remain after 1000 years.
You Find

202. You Finish & Round to the Nearest Hundredth
The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of grams, how much will remain after 1000 years.
You Finish & Round to the Nearest Hundredth

204. Link Back to Table of Contents
grams
Link Back to Table of Contents

205. Newton's
Law
of Cooling

206. is the Original temperature of the object

207. is the temperature of the surrounding air

208. is the final temperature of the object after t minutes

209. is the rate at which the object is cooling

210. When the soup is removed from the pot, it is 212ºF.
A chef wants the soup to be served to the customer at a temperature of no less than 160º F It has been determined that the cooling rate r for this soup is 0.21ºF per minute.
When the soup is removed from the pot, it is 212ºF.
If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.
To the nearest minute

211. When the soup is removed from the pot, it is 212ºF.
A chef wants the soup to be served to the customer at a temperature of no less than 160º F It has been determined that the cooling rate r for this soup is 0.21ºF per minute.
When the soup is removed from the pot, it is 212ºF.
If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

212. When the soup is removed from the pot, it is 212ºF.
A chef wants the soup to be served to the customer at a temperature of no less than 160º F It has been determined that the cooling rate r for this soup is 0.21ºF per minute.
When the soup is removed from the pot, it is 212ºF.
If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

213. When the soup is removed from the pot, it is 212ºF.
A chef wants the soup to be served to the customer at a temperature of no less than 160º F It has been determined that the cooling rate r for this soup is 0.21ºF per minute.
When the soup is removed from the pot, it is 212ºF.
If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

214. When the soup is removed from the pot, it is 212ºF.
A chef wants the soup to be served to the customer at a temperature of no less than 160º F It has been determined that the cooling rate r for this soup is 0.21ºF per minute.
When the soup is removed from the pot, it is 212ºF.
If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

215. When the soup is removed from the pot, it is 212ºF.
A chef wants the soup to be served to the customer at a temperature of no less than 160º F It has been determined that the cooling rate r for this soup is 0.21ºF per minute.
When the soup is removed from the pot, it is 212ºF.
If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

216. Isolate

217. Law #2
Law #3

218. Law #2
Law #3 1
Calculator Ready Form
To the nearest minute

219. Law #2
Law #3 1
Calculator Ready Form
To the nearest minute
2 min

220. You Try

221. To the nearest hundredth of a minute
Given that the original temperature of the coffee is 155ºF and the room temperature is 75ºF, determine after how many minutes the coffee will be 110ºF
To the nearest hundredth of a minute

222. To the nearest hundredth of a minute
Given that the original temperature of the coffee is 155ºF and the room temperature is 75ºF, determine after how many minutes the coffee will be 110ºF
To the nearest hundredth of a minute
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2.75 min