1. The document provides examples of solving exponential and logarithmic equations.
2. One example solves the equation log 4 (x + 3) = 2 and finds the solution is x = 13.
3. Another example solves the equation log 2 x + log 2 (x - 7) = 3 and finds the solutions are x = 8 or x = -1.
Ejercicios de algebra con pasos y una breve descripción del tema
como sumas y restas de polinomios, división sintética, entre otras.
Descripción de cada uno de los ejercicios propuestos.
Ejercicios de algebra con pasos y una breve descripción del tema
como sumas y restas de polinomios, división sintética, entre otras.
Descripción de cada uno de los ejercicios propuestos.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Under the Financial Crisis spreading all over the world, are you thinking about seizing this opportunity opening up your own business?
Then this one is definitely what you need for reminding yourself all the time!
A presentation held at the Norwegian National Open Access day in Bergen, September 26th, 2013. Describes what has led to increased open access at HiOA? Are there policies, incentive scheme, a combination of these? What results has it given? In addition, it describes the process that led to the current open access policy.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
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Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
2. 2. Exponential and Logarithmic Equations.
Exponential equations: An exponential equation is an equation
containing a variable in an exponent.
Logarithmic equations: Logarithmic equations contain
logarithmic expressions and constants.
Property of Logarithms, part 2:
≠ 1, then
If x, y and a are positive numbers, a
If x = y, then log a x = log a y
5. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
6. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3
7. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
8. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3
9. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
10. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2
11. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
12. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2
13. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.
14. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.
ln 3 − 2 ln 2
x=
ln 2 − 2 ln 3
15. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.
ln 3 − 2 ln 2
x= Divide both sides by ln2 - 2ln3
ln 2 − 2 ln 3
16. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
17. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2
4 = x+3
18. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
19. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3
20. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
21. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
13 = x
22. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
13 = x Solve for x.
23. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
13 = x Solve for x.
All solutions of Logarithmic equations must be checked,
because negative numbers do not have logarithms.
24. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
25. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3
26. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
27. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7)
28. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
29. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x
30. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
31. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2
0 = x − 7x − 8
32. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
33. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1)
34. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1) Solve by factoring
35. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1) Solve by factoring
x = 8 or x = -1
36. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1) Solve by factoring
x = 8 or x = -1 Check!
55. 2. Quiz 4.
1. How long does it take to double an investment of $ 20,000.00 in a
bank paying an interest rate of 4% per year compounded monthly?
Find the value of x in the following equations. Check your answers.
2. 3x+4 = e5 x
3. log12 ( x − 7 ) = 1− log12 ( x − 3)
4. Character that maintained a robust dispute with Newton over the
priority of invention of calculus.
5. How did Evariste Galois die, two days after leaving prison, at age
21?
6. Why isn’t there a Nobel Prize in mathematics?