Lesson 24: Solving Exponential Equations
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Lesson 24: Solving Exponential Equations Classwork Opening Exercise In Lesson 7, we modeled a population of bacteria that doubled every day by the function ππ(π‘π‘) = 2π‘π‘ , where π‘π‘ was the time in days. We wanted to know the value of π‘π‘ when there were 10 bacteria. Since we didnβt yet know about logarithms, we approximated the value of π‘π‘ numerically and we found that ππ(π‘π‘) = 10 at approximately π‘π‘ β 3.32 days. Use your knowledge of logarithms to find an exact value for π‘π‘ when ππ(π‘π‘) = 10, and then use your calculator to approximate that value to four decimal places.
Exercises 1.
Fiona modeled her data from the bean-flipping experiment in Lesson 23 by the function ππ(π‘π‘) = 1.263(1.357)π‘π‘ , and Gregor modeled his data with the function ππ(π‘π‘) = 0.972(1.629)π‘π‘ . a.
Without doing any calculating, determine which student, Fiona or Gregor, accumulated 100 beans first. Explain how you know.
b.
Using Fionaβs model β¦ i.
How many trials would be needed for her to accumulate 100 beans?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
ii.
c.
How many trials would be needed for her to accumulate 1,000 beans?
Using Gregorβs model β¦ i.
How many trials would be needed for him to accumulate 100 beans?
ii.
How many trials would be needed for him to accumulate 1,000 beans?
.
d.
Was your prediction in part (a) correct? If not, what was the error in your reasoning?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
2.
Fiona wants to know when her model ππ(π‘π‘) = 1.263(1.357)π‘π‘ predicts accumulations of 500, 5,000, and 50,000 beans, but she wants to find a way to figure it out without doing the same calculation three times. a.
Let the positive number ππ represent the number of beans that Fiona wants to have. Then solve the equation 1.263(1.357)π‘π‘ = ππ for π‘π‘.
b.
Your answer to part (a) can be written as a function ππ of the number of beans ππ, where ππ > 0. Explain what this function represents.
c.
When does Fionaβs model predict that she will accumulate β¦ i.
500 beans?
ii.
5000 beans?
iii.
50,000 beans?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
3.
Gregor states that the function ππ that he found to model his bean-flipping data can be written in the form ππ(π‘π‘) = 0.972οΏ½10log(1.629)π‘π‘ οΏ½. Since log(1.629) β 0.2119, he is using ππ(π‘π‘) = 0.972(100.2119π‘π‘ ) as his new model. a.
Is Gregor correct? Is ππ(π‘π‘) = 0.972οΏ½10log(1.629)π‘π‘ οΏ½ an equivalent form of his original function? Use properties of exponents and logarithms to explain how you know.
b.
Gregor also wants to find a function that will help him to calculate the number of trials his function ππ predicts it will take to accumulate 500, 5,000, and 50,000 beans. Let the positive number ππ represent the number of beans that Gregor wants to have. Solve the equation 0.972(100.2119π‘π‘ ) = ππ for π‘π‘.
c.
Your answer to part (b) can be written as a function ππ of the number of beans ππ, where ππ > 0. Explain what this function represents.
d.
When does Gregorβs model predict that he will accumulate β¦ i.
500 beans?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
ii.
5,000 beans?
.
iii.
4.
50,000 beans?
Helena and Karl each change the rules for the bean experiment. Helena started with four beans in her cup and added one bean for each that landed marked-side up for each trial. Karl started with one bean in his cup but added two beans for each that landed marked-side up for each trial. a.
Helena modeled her data by the function β(π‘π‘) = 4.127(1.468π‘π‘ ). Explain why her values of ππ = 4.127 and ππ = 1.468 are reasonable.
b.
Karl modeled his data by the function ππ(π‘π‘) = 0.897(1.992π‘π‘ ). Explain why his values of ππ = 0.897 and ππ = 1.992 are reasonable.
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
c.
At what value of π‘π‘ do Karl and Helena have the same number of beans?
d.
Use a graphing utility to graph π¦π¦ = β(π‘π‘) and π¦π¦ = ππ(π‘π‘) for 0 < π‘π‘ < 10.
e.
Explain the meaning of the intersection point of the two curves π¦π¦ = β(π‘π‘) and π¦π¦ = ππ(π‘π‘) in the context of this problem.
f.
Which student reaches 20 beans first? Does the reasoning you used with whether Gregor or Fiona would get to 100 beans first hold true here? Why or why not?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
For the following functions ππ and ππ, solve the equation ππ(π₯π₯) = ππ(π₯π₯). Express your solutions in terms of logarithms. 5.
ππ(π₯π₯) = 10(3.7)π₯π₯+1 , ππ(π₯π₯) = 5(7.4) π₯π₯
6.
ππ(π₯π₯) = 135(5)3π₯π₯+1 , ππ(π₯π₯) = 75(3)4β3π₯π₯
7.
ππ(π₯π₯) = 100π₯π₯
3 +π₯π₯ 2 β4π₯π₯
, ππ(π₯π₯) = 102π₯π₯
Lesson 24: Date:
2 β6π₯π₯
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
8.
ππ(π₯π₯) = 48οΏ½4π₯π₯
9.
ππ(π₯π₯) = ππ sin
2 +3π₯π₯
2 (π₯π₯)
οΏ½, ππ(π₯π₯) = 3οΏ½8π₯π₯
2 +4π₯π₯+4
οΏ½
2 (π₯π₯)
, ππ(π₯π₯) = ππ cos
10. ππ(π₯π₯) = (0.49)cos(π₯π₯)+sin(π₯π₯) , ππ(π₯π₯) = (0. 7)2 sin(π₯π₯)
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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S.167 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 24
ALGEBRA II
Problem Set 1.
Solve the following equations. a. b. c. d. e. f. g. h. i.
2.
3.
4. 5.
5 β 23π₯π₯+5 = 10240
43π₯π₯β1 = 32
3 β 25π₯π₯ = 216
5 β 113π₯π₯ = 120 7 β 9π₯π₯ = 5405
β3 β 33π₯π₯ = 9
log(400) β 85π₯π₯ = log(160000)
Jack came up with the model ππ(π‘π‘) = 1.033(1.707)π‘π‘ for the first bean activity. When does his model predict that he would have 50,000 beans? If instead of beans in the first bean activity you were using fair coins, when would you expect to have $1,000,000? Let ππ(π₯π₯) = 2 β 3π₯π₯ and ππ(π₯π₯) = 3 β 2π₯π₯ . b.
Which function is growing faster as π₯π₯ increases? Why?
When will ππ(π₯π₯) = ππ(π₯π₯)?
A population of E. coli bacteria can be modeled by the function πΈπΈ(π‘π‘) = 500(11.547)π‘π‘ , and a population of Salmonella bacteria can be modeled by the function ππ(π‘π‘) = 4000(3.668)π‘π‘ , where π‘π‘ measures time in hours. a.
b.
7.
3 β 62π₯π₯ = 648
Lucy came up with the model ππ(π‘π‘) = 0.701(1.382)π‘π‘ for the first bean activity. When does her model predict that she would have 1,000 beans?
a.
6.
2 β 5π₯π₯+3 = 6250
Graph these two functions on the same set of axes. At which value of π‘π‘ does it appear that the graphs intersect?
Use properties of logarithms to find the time π‘π‘ when these two populations are the same size. Give your answer to two decimal places.
Chain emails contain a message suggesting you will have bad luck if you do not forward the email to others. Suppose a student started a chain email by sending the message to 10 friends and asking those friends to each send the same email to 3 more friends exactly one day after receiving the message. Assuming that everyone that gets th the email participates in the chain, we can model the number of people who will receive the email on the ππ day by the formula πΈπΈ(ππ) = 10(3ππ ), where ππ = 0 indicates the day the original email was sent. P
a.
b.
If we assume the population of the United States is 318 million people and everyone who receives the email sends it to 3 people who have not received it previously, how many days until there are as many emails being sent out as there are people in the United States? The population of Earth is approximately 7.1 billion people. On what day will 7.1 billion emails be sent out? Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
8.
Solve the following exponential equations. a. b. c. d.
9.
10(3π₯π₯β5) = 7π₯π₯ π₯π₯
35 = 24π₯π₯β2 10π₯π₯ 4π₯π₯
2 +5
= 1002π₯π₯
2 β3π₯π₯+4
2 +π₯π₯+2
= 25π₯π₯β4
Solve the following exponential equations. a. b.
(2π₯π₯ )π₯π₯ = 8π₯π₯ (3π₯π₯ )π₯π₯ = 12
10. Solve the following exponential equations. a. b.
10π₯π₯+1 β 10π₯π₯β1 = 1287 2(4π₯π₯ ) + 4π₯π₯+1 = 342
11. Solve the following exponential equations. a. b.
c. d. e.
(10π₯π₯ )2 β 3(10π₯π₯ ) + 2 = 0
Hint: Let π’π’ = 10π₯π₯ and solve for π’π’ before solving for π₯π₯.
(2π₯π₯ )2 β 3(2π₯π₯ ) β 4 = 0
3(ππ π₯π₯ )2 β 8(ππ π₯π₯ ) β 3 = 0 4π₯π₯ + 7(2π₯π₯ ) + 12 = 0
(10π₯π₯ )2 β 2(10π₯π₯ ) β 1 = 0
12. Solve the following systems of equations. a. b. c.
2π₯π₯+2π¦π¦ = 8 42π₯π₯+π¦π¦ = 1
22π₯π₯+π¦π¦β1 = 32 4π₯π₯β2π¦π¦ = 2 23π₯π₯ = 82π¦π¦+1 92π¦π¦ = 33π₯π₯β9
13. Because ππ(π₯π₯) = log ππ (π₯π₯) is an increasing function, we know that if ππ < ππ, then log ππ (ππ) < log ππ (ππ). Thus, if we take logarithms of both sides of an inequality, then the inequality is preserved. Use this property to solve the following inequalities. a. b. c. d. e.
4π₯π₯ > 2 π₯π₯
5 3
οΏ½ οΏ½ >9 7
4π₯π₯ > 8π₯π₯β1
3π₯π₯+2 > 53β2π₯π₯ 3 π₯π₯
4 π₯π₯+1
οΏ½ οΏ½ >οΏ½ οΏ½ 4
3
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Lesson 24: Solving Exponential Equations Classwork Opening Exercise In Lesson 7, we modeled a population of bacteria that doubled every day by the function ππ(π‘π‘) = 2π‘π‘ , where π‘π‘ was the time in days. We wanted to know the value of π‘π‘ when there were 10 bacteria. Since we didnβt yet know about logarithms, we approximated the value of π‘π‘ numerically and we found that ππ(π‘π‘) = 10 at approximately π‘π‘ β 3.32 days. Use your knowledge of logarithms to find an exact value for π‘π‘ when ππ(π‘π‘) = 10, and then use your calculator to approximate that value to four decimal places.
Exercises 1.
Fiona modeled her data from the bean-flipping experiment in Lesson 23 by the function ππ(π‘π‘) = 1.263(1.357)π‘π‘ , and Gregor modeled his data with the function ππ(π‘π‘) = 0.972(1.629)π‘π‘ . a.
Without doing any calculating, determine which student, Fiona or Gregor, accumulated 100 beans first. Explain how you know.
b.
Using Fionaβs model β¦ i.
How many trials would be needed for her to accumulate 100 beans?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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S.160 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
ii.
c.
How many trials would be needed for her to accumulate 1,000 beans?
Using Gregorβs model β¦ i.
How many trials would be needed for him to accumulate 100 beans?
ii.
How many trials would be needed for him to accumulate 1,000 beans?
.
d.
Was your prediction in part (a) correct? If not, what was the error in your reasoning?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
2.
Fiona wants to know when her model ππ(π‘π‘) = 1.263(1.357)π‘π‘ predicts accumulations of 500, 5,000, and 50,000 beans, but she wants to find a way to figure it out without doing the same calculation three times. a.
Let the positive number ππ represent the number of beans that Fiona wants to have. Then solve the equation 1.263(1.357)π‘π‘ = ππ for π‘π‘.
b.
Your answer to part (a) can be written as a function ππ of the number of beans ππ, where ππ > 0. Explain what this function represents.
c.
When does Fionaβs model predict that she will accumulate β¦ i.
500 beans?
ii.
5000 beans?
iii.
50,000 beans?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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S.162 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
3.
Gregor states that the function ππ that he found to model his bean-flipping data can be written in the form ππ(π‘π‘) = 0.972οΏ½10log(1.629)π‘π‘ οΏ½. Since log(1.629) β 0.2119, he is using ππ(π‘π‘) = 0.972(100.2119π‘π‘ ) as his new model. a.
Is Gregor correct? Is ππ(π‘π‘) = 0.972οΏ½10log(1.629)π‘π‘ οΏ½ an equivalent form of his original function? Use properties of exponents and logarithms to explain how you know.
b.
Gregor also wants to find a function that will help him to calculate the number of trials his function ππ predicts it will take to accumulate 500, 5,000, and 50,000 beans. Let the positive number ππ represent the number of beans that Gregor wants to have. Solve the equation 0.972(100.2119π‘π‘ ) = ππ for π‘π‘.
c.
Your answer to part (b) can be written as a function ππ of the number of beans ππ, where ππ > 0. Explain what this function represents.
d.
When does Gregorβs model predict that he will accumulate β¦ i.
500 beans?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
ii.
5,000 beans?
.
iii.
4.
50,000 beans?
Helena and Karl each change the rules for the bean experiment. Helena started with four beans in her cup and added one bean for each that landed marked-side up for each trial. Karl started with one bean in his cup but added two beans for each that landed marked-side up for each trial. a.
Helena modeled her data by the function β(π‘π‘) = 4.127(1.468π‘π‘ ). Explain why her values of ππ = 4.127 and ππ = 1.468 are reasonable.
b.
Karl modeled his data by the function ππ(π‘π‘) = 0.897(1.992π‘π‘ ). Explain why his values of ππ = 0.897 and ππ = 1.992 are reasonable.
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
c.
At what value of π‘π‘ do Karl and Helena have the same number of beans?
d.
Use a graphing utility to graph π¦π¦ = β(π‘π‘) and π¦π¦ = ππ(π‘π‘) for 0 < π‘π‘ < 10.
e.
Explain the meaning of the intersection point of the two curves π¦π¦ = β(π‘π‘) and π¦π¦ = ππ(π‘π‘) in the context of this problem.
f.
Which student reaches 20 beans first? Does the reasoning you used with whether Gregor or Fiona would get to 100 beans first hold true here? Why or why not?
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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S.165 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
For the following functions ππ and ππ, solve the equation ππ(π₯π₯) = ππ(π₯π₯). Express your solutions in terms of logarithms. 5.
ππ(π₯π₯) = 10(3.7)π₯π₯+1 , ππ(π₯π₯) = 5(7.4) π₯π₯
6.
ππ(π₯π₯) = 135(5)3π₯π₯+1 , ππ(π₯π₯) = 75(3)4β3π₯π₯
7.
ππ(π₯π₯) = 100π₯π₯
3 +π₯π₯ 2 β4π₯π₯
, ππ(π₯π₯) = 102π₯π₯
Lesson 24: Date:
2 β6π₯π₯
Solving Exponential Equations 10/29/14
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S.166 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
8.
ππ(π₯π₯) = 48οΏ½4π₯π₯
9.
ππ(π₯π₯) = ππ sin
2 +3π₯π₯
2 (π₯π₯)
οΏ½, ππ(π₯π₯) = 3οΏ½8π₯π₯
2 +4π₯π₯+4
οΏ½
2 (π₯π₯)
, ππ(π₯π₯) = ππ cos
10. ππ(π₯π₯) = (0.49)cos(π₯π₯)+sin(π₯π₯) , ππ(π₯π₯) = (0. 7)2 sin(π₯π₯)
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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S.167 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 24
ALGEBRA II
Problem Set 1.
Solve the following equations. a. b. c. d. e. f. g. h. i.
2.
3.
4. 5.
5 β 23π₯π₯+5 = 10240
43π₯π₯β1 = 32
3 β 25π₯π₯ = 216
5 β 113π₯π₯ = 120 7 β 9π₯π₯ = 5405
β3 β 33π₯π₯ = 9
log(400) β 85π₯π₯ = log(160000)
Jack came up with the model ππ(π‘π‘) = 1.033(1.707)π‘π‘ for the first bean activity. When does his model predict that he would have 50,000 beans? If instead of beans in the first bean activity you were using fair coins, when would you expect to have $1,000,000? Let ππ(π₯π₯) = 2 β 3π₯π₯ and ππ(π₯π₯) = 3 β 2π₯π₯ . b.
Which function is growing faster as π₯π₯ increases? Why?
When will ππ(π₯π₯) = ππ(π₯π₯)?
A population of E. coli bacteria can be modeled by the function πΈπΈ(π‘π‘) = 500(11.547)π‘π‘ , and a population of Salmonella bacteria can be modeled by the function ππ(π‘π‘) = 4000(3.668)π‘π‘ , where π‘π‘ measures time in hours. a.
b.
7.
3 β 62π₯π₯ = 648
Lucy came up with the model ππ(π‘π‘) = 0.701(1.382)π‘π‘ for the first bean activity. When does her model predict that she would have 1,000 beans?
a.
6.
2 β 5π₯π₯+3 = 6250
Graph these two functions on the same set of axes. At which value of π‘π‘ does it appear that the graphs intersect?
Use properties of logarithms to find the time π‘π‘ when these two populations are the same size. Give your answer to two decimal places.
Chain emails contain a message suggesting you will have bad luck if you do not forward the email to others. Suppose a student started a chain email by sending the message to 10 friends and asking those friends to each send the same email to 3 more friends exactly one day after receiving the message. Assuming that everyone that gets th the email participates in the chain, we can model the number of people who will receive the email on the ππ day by the formula πΈπΈ(ππ) = 10(3ππ ), where ππ = 0 indicates the day the original email was sent. P
a.
b.
If we assume the population of the United States is 318 million people and everyone who receives the email sends it to 3 people who have not received it previously, how many days until there are as many emails being sent out as there are people in the United States? The population of Earth is approximately 7.1 billion people. On what day will 7.1 billion emails be sent out? Lesson 24: Date:
Solving Exponential Equations 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.168 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M3
ALGEBRA II
8.
Solve the following exponential equations. a. b. c. d.
9.
10(3π₯π₯β5) = 7π₯π₯ π₯π₯
35 = 24π₯π₯β2 10π₯π₯ 4π₯π₯
2 +5
= 1002π₯π₯
2 β3π₯π₯+4
2 +π₯π₯+2
= 25π₯π₯β4
Solve the following exponential equations. a. b.
(2π₯π₯ )π₯π₯ = 8π₯π₯ (3π₯π₯ )π₯π₯ = 12
10. Solve the following exponential equations. a. b.
10π₯π₯+1 β 10π₯π₯β1 = 1287 2(4π₯π₯ ) + 4π₯π₯+1 = 342
11. Solve the following exponential equations. a. b.
c. d. e.
(10π₯π₯ )2 β 3(10π₯π₯ ) + 2 = 0
Hint: Let π’π’ = 10π₯π₯ and solve for π’π’ before solving for π₯π₯.
(2π₯π₯ )2 β 3(2π₯π₯ ) β 4 = 0
3(ππ π₯π₯ )2 β 8(ππ π₯π₯ ) β 3 = 0 4π₯π₯ + 7(2π₯π₯ ) + 12 = 0
(10π₯π₯ )2 β 2(10π₯π₯ ) β 1 = 0
12. Solve the following systems of equations. a. b. c.
2π₯π₯+2π¦π¦ = 8 42π₯π₯+π¦π¦ = 1
22π₯π₯+π¦π¦β1 = 32 4π₯π₯β2π¦π¦ = 2 23π₯π₯ = 82π¦π¦+1 92π¦π¦ = 33π₯π₯β9
13. Because ππ(π₯π₯) = log ππ (π₯π₯) is an increasing function, we know that if ππ < ππ, then log ππ (ππ) < log ππ (ππ). Thus, if we take logarithms of both sides of an inequality, then the inequality is preserved. Use this property to solve the following inequalities. a. b. c. d. e.
4π₯π₯ > 2 π₯π₯
5 3
οΏ½ οΏ½ >9 7
4π₯π₯ > 8π₯π₯β1
3π₯π₯+2 > 53β2π₯π₯ 3 π₯π₯
4 π₯π₯+1
οΏ½ οΏ½ >οΏ½ οΏ½ 4
3
Lesson 24: Date:
Solving Exponential Equations 10/29/14
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