Unformatted text preview: Lesson 11.1 Skills Practice Name Date All the Pieces of the Puzzle
Exponential and Logarithmic Forms
Vocabulary 1. Give an example of a logarithmic equation. Problem Set
Arrange the given terms to create a true exponential equation and a true logarithmic equation. 2. 3, 11, 1331 1. 25, 5, 2
Exponential equation: 5 5 25
2 Logarithmic equation: log525 5 2 11 4. 1296, 4, 6 5. 23, ____
1 , 8
512 1 , 4, 16 6. __
2 7. 17, ____
1 , 22
289 8. __
1 , 9, 729
3 © Carnegie Learning 3. 2, 256, 8 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 861 861 28/11/13 4:31 PM Lesson 11.1 Skills Practice page 2 Solve for the unknown. 9. log7343 5 n
log7343 5 n 10. log__164 5 n
4 n 5 343
7
n 5 73
7
n 5 3 ( ) 11 logn1024 5 5 12. logn ____
1 5 24
625 3 )5 __
4 8
13. logn (
4 14. log n 5 6 15. log416 5 n 16. log81__
1 5 n
9 862 ( ) © Carnegie Learning 11 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 862 28/11/13 4:31 PM Lesson 11.1 Skills Practice Name page 3 Date Estimate the logarithm to the tenths place. Explain how you determined your answer. 17. log248
The logarithm log248 is approximately equal to 5.5.
log232 , log248 , log264
5 , n , 6 Because 48 is halfway between 32 and 64, the approximation should be halfway between
5 and 6. 18. log32 11
( ____ ) © Carnegie Learning 19. log8 1
495 20. log653 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 863 863 28/11/13 4:31 PM Lesson 11.1 Skills Practice page 4 21. log96000 ( ) 22. log7___
1
10 ( ) 23. log__1___
1
24
2 24. log__40.85
3 864 © Carnegie Learning 11 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 864 28/11/13 4:31 PM Lesson 11.1 Skills Practice Name page 5 Date Determine the appropriate base of each logarithm. Explain your reasoning. 25. logb29 5 2.1
log525 , logb29 , log5125
2 , 2.1 , 3
Because the value of the exponent is 2.1, that means that the argument 29 should be close to its
lower limit. When the base is 5, 29 is very close to the lower limit of 25, whereas when the base is
4, 29 is not very close to the lower limit of 16. © Carnegie Learning 26. logb35 5 1.9 11 ( ) 27. logb__
1 5 21.9
7 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 865 865 28/11/13 4:31 PM Lesson 11.1 Skills Practice page 6 28. logb80 5 3.9 29. logb6 5 0.9 11 © Carnegie Learning 30. logb66 5 3.1 866 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 866 28/11/13 4:31 PM Lesson 11.2 Skills Practice Name Date Mad Props
Properties of Logarithms
Problem Set
Rewrite each logarithmic expression in expanded form using the properties of logarithms. 1. log3( 5x )
log3(5x) 5 log35 1 log3x ( ) 3. log7( n4 ) 4. log __
x
7 5. log2( mn ) 6. log ( pq ) 7. ln ( x2 ) 8. ln __
c
3 9. log3( 7x2 ) 10. ln ( 2x3y2 ) ( ) xy
11. log ___
5
© Carnegie Learning ( ) 2. log5__
a
b ( ) 11 ( ) ( ) 4 12. log7___
3yx
( ) x 13. ln ___
7y 2 14. log5___
7x3
y 15. log ( xyz ) 16. ln _______
x 1 1
(y 1 3)2 ( ) Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 867 867 28/11/13 4:31 PM Lesson 11.2 Skills Practice page 2 Rewrite each logarithmic expression as a single logarithm. 17. log x 2 2 log y log x 2 2 log y 5 log __
( yx ) 18. 3 log4x 1 log4y 2 log4z 2 19. 6 log2x 2 2 log2x 20. log 3 1 2 log 7 2 log 6 21. log x 1 3 log y 2 __
1 log z
2 22. 7 log3x 2 (2 log3x 1 5 log3y) 23. 2 ln (2x 1 3) 2 4 ln (y 2 2) 24. ln (x 2 7) 2 2(ln x 1 ln y) 11
Suppose w 5 logb2, x 5 logb3, y 5 logb7, and z 5 logb11. Write an algebraic expression for each
logarithmic expression. 26. logb98 25. logb33
logb33 5 logb(3 ? 11) 5 logb3 1 logb11 ( ) 868 ( ) 2 27. logb__
3 28. logb__
7
8 29. logb1.5 30. logb2.75 © Carnegie Learning 5 x 1 z Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 868 28/11/13 4:31 PM Lesson 11.3 Skills Practice Name Date What’s Your Strategy? Solving Exponential Equations
Vocabulary 1. Define the Change of Base Formula and explain how it is used. Problem Set
Solve each exponential equation by using the Change of Base Formula. 1. 6x255 24 2. 12x145 65 6x25
5 24
x 2 5 5 log624 11 ______ log 24
x 2 5 5
log 6
x 2 5 ¯ 1.774
x ¯ 6.774 4. 8
5x
5 71 © Carnegie Learning 3. 7
3x
5 15 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 869 869 28/11/13 4:31 PM Lesson 11.3 Skills Practice 5. 4
x13
2 7 5 32 ( ) page 2 6. 11x241 8 5 59 ( ) 7. 4__
2 5 248
3 8. 9__
3 5 999
5 9. 2(5)2x111 4 5 18 10. 28(2)x2925 5 277 3x 2x © Carnegie Learning 11 870 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 870 28/11/13 4:31 PM Lesson 11.3 Skills Practice page 3 Name Date Solve each exponential equation using properties of logarithms. 12. 15x125 60 11. 7
x22
5 28
x22
7
5 28
(x 2 2) log 7 5 log 28 ______ log 28
x 2 2 5
log 7
x 2 2 ¯ 1.712
x ¯ 3.712 13. 5
4x5 32 14. 9
3x5 124 11 16. 14x261 3 5 73 © Carnegie Learning 15. 3x172 5 5 63 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 871 871 28/11/13 4:31 PM Lesson 11.3 Skills Practice ( ) 11 page 4 ( ) 2x
4 5 348 17. 6__
7 4x 18. 8 __
5 5 448
8 19. 3(4)3x261 2 5 35 20. 27(3)x162 8 5 2162 © Carnegie Learning 872 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 872 28/11/13 4:31 PM Lesson 11.3 Skills Practice page 5 Name Date Solve each exponential equation. Explain why you chose the method that you used. 21. 11x245 343 11x24 5 343 (x 2 4) log 11 5 log 343
log 343
x 2 4 5
log 11 x 2 4 ¯ 2.435 _______ x ¯ 6.435 I took the log of both sides, because 343 cannot be written as a power of 11. 22. 5
2x+15 3125 11 ( ) © Carnegie Learning 5x 23. 16__
8 5 752
9 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 873 873 28/11/13 4:31 PM Lesson 11.3 Skills Practice page 6 24. 235x5 736 ( ) 4x 25. ___
7 5 49
12 11 © Carnegie Learning 26. 145x2122 8 5 2736 874 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 874 28/11/13 4:31 PM Lesson 11.4 Skills Practice Name Date Logging On
Solving Logarithmic Equations
Problem Set
Solve each logarithmic equation. Check your answer(s). 1. log2(x2 2 x) 5 1 2. log15(x2 2 2x) 5 1 log2(x2 2 x) 5 1
1 5 x2 2 x
2
0 5 x2 2 x 2 2
0 5 (x 1 1)(x 2 2)
x 5 21, 2
Check:
log2( (21)22 (21) ) 0 1 11 log2(1 1 1) 0 1
log22 5 1
log2(22 2 2) 0 1
log2(4 2 2) 0 1 © Carnegie Learning log22 5 1 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 875 875 28/11/13 4:31 PM Lesson 11.4 Skills Practice page 2 3. log6(x2 1 5x) 5 2 4. log2(x2 1 6x) 5 4 5. log4(x2 2 12x) 5 3 6. log10(x2 1 15x) 5 2 © Carnegie Learning 11 876 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 876 28/11/13 4:31 PM Lesson 11.4 Skills Practice page 3 Name 7. log3(3x2118x) 5 4 Date 8. log4(2x22 28x) 5 3 © Carnegie Learning 11 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 877 877 28/11/13 4:31 PM Lesson 11.4 Skills Practice page 4 Use the properties of logarithms to solve each logarithmic equation. Check your answer(s). 9. 2 log3x 2 log38 5 log3(x 2 2) 10. log4(x 1 3) 1 log4x 5 1 2 log3x 2 log38 5 log3(x 2 2)
log3x22 log38 5 log3(x 2 2)
2
log3 x 5 log3(x 2 2)
8 ( __ )
__ x 5 x 2 2
8
2 x 2 5 8x 2 16
x2 2 8x 1 16 5 0
(x 2 4)2 5 0
x 5 4
Check:
2 log34 2 log38 0 log3(4 2 2)
log316 2 log38 0 log32
log3 16 0 log32
8
log32 5 log32 ( ___ ) © Carnegie Learning 11 878 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 878 28/11/13 4:31 PM Lesson 11.4 Skills Practice page 5 Name 11. log (2x2 1 3) 1 log 2 5 log 10x Date 12. log2x 1 log2(x 2 6) 5 4 © Carnegie Learning 11 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 879 879 28/11/13 4:31 PM Lesson 11.4 Skills Practice 13. 2 log5x 2 log54 5 log5(8 2 x) page 6 14. log23 1 log2(3x21 4) 5 log2( 39x ) © Carnegie Learning 11 880 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 880 28/11/13 4:31 PM Lesson 11.4 Skills Practice page 7 Name Date ( ) 15. ln x2 1 ___
15 1 ln 2 5 ln (11x)
2 ( ) ( ) 16. log4__
1 x2 2 6 2 log4__
1 5 log4x
5
5 © Carnegie Learning 11 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 881 881 28/11/13 4:31 PM © Carnegie Learning 11 882 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 882 28/11/13 4:31 PM Lesson 11.5 Skills Practice Name Date So When Will I Use This?
Applications of Exponential and Logarithmic Equations
Problem Set
The amount of a radioactive isotope remaining can be modeled using the formula A 5 A
0e2kt, where
t represents the time in years, A represents the amount of the isotope remaining in grams after t years,
A0 represents the original amount of the isotope in grams, and k is the decay constant. Use this formula to
solve each problem. 1. Strontium90 is a radioactive isotope with a halflife of about 29 years. Calculate the decay constant
for Strontium90. Then find the amount of 100 grams of Strontium90 remaining after 120 years. A 5 A0 e2kt 1 A0 5 A0 e2k(29)
2
1 5 e229k
2 ln 1 5 229k
2 __ __
( __ ) 11 k ¯ 0.0239 The decay constant for Strontium90 is about 0.0239.
A 5 100e20.0239(120)
A ¯ 5.681
After 120 years, there would be about 5.681 grams remaining. © Carnegie Learning 2. Radium226 is a radioactive isotope with a halflife of about 1622 years. Calculate the decay
constant for Radium226. Then find the amount of 20 grams of Radium226 remaining after
500 years. Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 883 883 28/11/13 4:31 PM Lesson 11.5 Skills Practice page 2 3. Carbon14 is a radioactive isotope with a halflife of about 5730 years. Calculate the decay constant
for Carbon14. Then find the amount of 6 grams of Carbon14 that will remain after 22,000 years. 4. Cesium137 is a radioactive isotope with a halflife of about 30 years. Calculate the decay constant
for Cesium137. Then calculate the percentage of a Cesium137 sample remaining after 100 years. © Carnegie Learning 11 884 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 884 28/11/13 4:31 PM Lesson 11.5 Skills Practice Name page 3 Date 5. Uranium232 is a radioactive isotope with a halflife of about 69 years. Calculate the decay constant
for Uranium232. Then calculate the percentage of a Uranium232 sample remaining after 200 years. 11 © Carnegie Learning 6. Rubidium87 is a radioactive isotope with a halflife of about 4.7 3 107years. Calculate the decay
constant for Rubidium87. Then calculate the percentage of a Rubidium87 sample remaining after
1,000,000 years. Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 885 885 28/11/13 4:31 PM Lesson 11.5 Skills Practice page 4 Use the given exponential equation to answer each question. Show your work. 7. The number of students exposed to the measles at a school can be modeled by the equation
S 5 10e0.15t
, where S represents the number of students exposed after t days. How many students
were exposed after eight days? S 5 10e0.15t
5 10e(0.15 ? 8)
5 10e1.2 ¯ 33.20116923
Approximately 33 students were exposed after eight days. 8. The minnow population in White Mountain Lake each year can be modeled by the equation
M 5 700(100.2t) , where M represents the minnow population t years from now. What will the minnow
population be in 15 years? 9. Aiden invested $600 in a savings account with continuous compound interest. The equation
V 5 600e0.05t
can be used to predict the value, V, of Aiden’s account after t years. What would the
value of Aiden’s account be after five years? 886 © Carnegie Learning 11 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 886 28/11/13 4:31 PM Lesson 11.5 Skills Practice Name page 5 Date 10. The rabbit population on Hare Island can be modeled by the equation R 5 60e0.09t, where
R represents the rabbit population t years from now. How many years from now will the rabbit
population of Hare Island be 177 rabbits? 11 © Carnegie Learning 11. A disease is destroying the elm tree population in the Dutch Forest. The equation N 5 16(100.15t) can
be used to predict the number of elm trees, N, killed by the disease t years from now. In how many
years from now will 406 elm trees have been killed by the disease? Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 887 887 28/11/13 4:31 PM Lesson 11.5 Skills Practice page 6 12. Manuel invested money in a savings account with continuous compound interest. The equation
V 5 10,000e0.03t
can be used to determine the value, V, of the account after t years. In how many
years will the value of the account be $12,000? ( ) 11 Use the formula M 5 log __
I , where M is the magnitude of an earthquake on the Richter scale, I0
I0
represents the intensity of a zerolevel earthquake the same distance from the epicenter, and I is the
number of times more intense an earthquake is than a zerolevel earthquake, to solve each problem. A
zerolevel earthquake has a seismographic reading of 0.001 millimeter at a distance of 100 kilometers
from the center. 13. An earthquake southwest of Chattanooga, Tennessee in 2003 had a seismographic reading of
79.43 millimeters registered 100 kilometers from the center. What was the magnitude of the
Tennessee earthquake of 2003 on the Richter scale? ( __ ) M 5 log I
I0 ( ______ ) M 5 log 79.43
0.001
M ¯ 4.9 14. An earthquake in Illinois in 2008 had a seismographic reading of 158.5 millimeters registered
100 kilometers from the center. What was the magnitude of the Illinois earthquake of 2008 on the
Richter scale? 888 © Carnegie Learning The Tennessee earthquake of 2003 measured 4.9 on the Richter scale. Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 888 28/11/13 4:31 PM Lesson 11.5 Skills Practice Name page 7 Date 15. An earthquake off the northern coast of California in 2005 had a seismographic reading of
15,849 millimeters registered 100 kilometers from the center. What was the magnitude of the
California earthquake in 2005 on the Richter scale? 16. The devastating earthquake in Haiti in 2010 had a magnitude of 7.0 on the Richter scale. What was
its seismographic reading in millimeters 100 kilometers from the center? © Carnegie Learning 11 17. Calculate the value of the seismographic reading for an earthquake of magnitude 6.4 on the
Richter scale. Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 889 889 28/11/13 4:31 PM Lesson 11.5 Skills Practice page 8 18. Calculate the value of the seismographic reading for an earthquake of magnitude 8.1 on the
Richter scale. Use the given formula to solve each problem. 19. The formula for the population of a species is n 5 k log (A), where n represents the population of a
species, A is the area of the region in which the species lives, and k is a constant that is determined
by field studies. Based on population samples, an area that is 1000 square miles contains
360 wolves. Calculate the value of k. Then use the formula to find the number of wolves remaining in
15 years if only 300 square miles of this area is still inhabitable. 11 n 5 k log (A) 360 5 k log 1000 k 5 120
The value of k is 120. n 5 120 log (A) 5 120 log 300 ¯ 297 20. The formula for the population of a species is n 5 k log (A), where n represents the population of a
species, A is the area of the region in which the species lives, and k is a constant that is determined
by field studies. Based on population samples, a rainforest that is 100 square miles contains
342 monkeys. Calculate the value of k. Then use the formula to find the number of monkeys remaining
in 5 years if only 40 square miles of the rainforest survives due to the current level of deforestation. 890 © Carnegie Learning In 15 years, there will be approximately 297 wolves remaining in the area. Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 890 28/11/13 4:31 PM Lesson 11.5 Skills Practice Name page 9 Date 21. The formula y 5 a 1 b ln t, where t represents the time in hours, y represents the amount of fresh
water produced in t hours, a represents the amount of fresh water produced in one hour, and b is the
rate of production, models the amount of fresh water produced from salt water during a desalinization
process. In one desalination plant, 15.26 cubic yards of fresh water can be produced in one hour with
a rate of production of 31.2. How much fresh water can be produced after 8 hours? 11 © Carnegie Learning 22. The formula y 5 a 1 b ln t, where t represents the time in hours, y represents the amount of fresh
water produced in t hours, a represents the amount of fresh water produced in one hour, and b is the
rate of production, models the amount of fresh water produced from salt water during a desalinization
process. At a desalination plant, 18.65 cubic yards of fresh water can be produced in one hour with a
rate of production of 34.5. How long will it take for the plant to produce 250 cubic yards of
fresh water? Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 891 891 28/11/13 4:31 PM Lesson 11.5 Skills Practice page 10 23. The relationship between the age of an item in years and its value is given by the equation
V
log __
C
__________
t5
, where t represents the age of the item in years, V represents the value of the item
log (1 2 r)
after t years, C represents the original value of the item, and r represents the yearly rate of
appreciation expressed as a decimal. A luxury car was originally purchased for $110,250 and is
currently valued at $65,200. The average rate of depreciate for this car is 10.3% per year. How old is
the car to the nearest tenth of a year? ( ) 11 24. The relationship between the age of an item in years and its value is given by the equation
V
log __
C
__________
t5
, where t represents the age of the item in years, V represents the value of the item after
log (1 2 r)
t years, C represents the original value of the item, and r represents the yearly rate of appreciation
expressed as a decimal. A 4year old car was originally purchased for $35,210. Its current value is
$16,394. What is this car’s annual rate of depreciation to the nearest tenth? © Carnegie Learning ( ) 892 Chapter 11 Skills Practice 504371_Skills_CH11_861892.indd 892 28/11/13 4:31 PM ...
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 Spring '16
 KAREN L SMALLEY
 Natural logarithm, Logarithm, Carnegie Learning

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