This document provides information about logarithmic and exponential forms including:
- Examples of logarithmic expressions and their exponential equivalents
- Rules for evaluating logarithmic expressions
- Examples of using logarithms to solve equations
- How to use logarithms to find the x-coordinate when given the y-coordinate on a curve
2 can be expressed as a power of 3, 3 can be expressed as a power of 5. Do not live in the world that has only rational numbers; it is irrational to do that.
2 can be expressed as a power of 3, 3 can be expressed as a power of 5. Do not live in the world that has only rational numbers; it is irrational to do that.
Apresentação sobre o algoritmo de ordenação Counting Sort, apresentada no 2º semestre de 2014, como um dos requisitos da disciplina de Projeto e Análise de Algoritmos, no Mestrado em Ciência da Computação, pela Universidade Federal de Lavras (UFLA). Disciplina ministrada pelo professor Dr. Sanderson L. Gonzaga de Oliveira.
Apresentação sobre o algoritmo de ordenação Counting Sort, apresentada no 2º semestre de 2014, como um dos requisitos da disciplina de Projeto e Análise de Algoritmos, no Mestrado em Ciência da Computação, pela Universidade Federal de Lavras (UFLA). Disciplina ministrada pelo professor Dr. Sanderson L. Gonzaga de Oliveira.
QUESTION 11. Find the limit.Does not exist-∞.docxmakdul
QUESTION 1
1. Find the limit.
Does not exist
-∞
5
∞
2 points
QUESTION 2
1. Express as a single logarithm.
(logat - logas) + 5logau
loga
logatu5s
loga
loga
2 points
QUESTION 3
1. Express as a single logarithm.
3log6x + 5log6(x - 6)
15 log6x(x - 6)
log6x(x - 6)
log6x3(x - 6)5
log6x(x - 6)15
2 points
QUESTION 4
1. Solve the equation.
x =
{-4}
{4}
2 points
QUESTION 5
1. Write as the sum and/or difference of logarithms. Express powers as factors.
log4
log4(x + 6) - 3log4x
log4(x + 6) -log4x
log4(x + 6) + 3log4x
3log4x -log4(x + 6)
2 points
QUESTION 6
1. Find the limit.
Does not exist
0
∞
-∞
2 points
QUESTION 7
1. Find the limit if it exists.
62
-
-62
2 points
QUESTION 8
1. Express as a single logarithm.
6logam - logan + logaj - 5logak
loga
loga
loga
loga
2 points
QUESTION 9
1. Find the limit.
1
∞
0
-∞
2 points
QUESTION 10
1. Write as the sum and/or difference of logarithms. Express powers as factors.
logw
logw11 +logwx - logw4
logw7x
logw11x -logw4
logw11 +logwx + logw4
2 points
QUESTION 11
1. Express as a single logarithm.
logcx + logcy
logcx ∙ logcy
logc
logcxy
logc(xy)
2 points
QUESTION 12
1. Find the limit.
-∞
-1
∞
0
2 points
QUESTION 13
1. Find the domain of the function.
h(x) =
{x|x ≠ -8, 0, 8}
all real numbers
{x|x ≠ 2}
{x|x ≠ 0}
2 points
QUESTION 14
1. Find the limit.
-∞
1
3
0
2 points
QUESTION 15
1. Find the intervals on which the function is continuous.
y =
continuous on the interval
continuous on the interval
continuous on the interval
continuous on the interval
2 points
QUESTION 16
1. Find the limit.
- 1/3
-∞
5
∞
2 points
QUESTION 17
1. Write as the sum and/or difference of logarithms. Express powers as factors.
log14 19/17
log1419 ÷ log1417
log1419 - log1417
log1417 - log1419
log1419 + log1419
2 points
QUESTION 18
1. Find the domain of the function.
g(x) =
{x|x ≠ 0}
{x|x ≠ -7, 7}
{x|x > 49}
all real numbers
2 points
QUESTION 19
1. Find the limit if it exists.
x3/4
64
3/4
256
192
2 points
QUESTION 20
1. Find the limit.
1
0
- 1/5
does not exist
2 points
QUESTION 21
1. Find the intervals on which the function is continuous.
y =
continuous on the intervals (-∞, -] and [, ∞)
continuous on the interval [, ∞)
continuous everywhere
continuous on the interval [-, ]
2 points
QUESTION 22
1. Find the intervals on which the function is continuous.
y =
continuous on the interval
continuous on the interval
continuous on the interval
continuous on the interval
2 points
QUESTION 23
1. Express as a single logarithm.
6logbq - logbr
logb
logbq6 ÷ logbr
logb
logb(q6 - r)
2 points
QUESTION 24
1. Find the intervals on which the function is continuous.
y =
discontinuous only when θ = 0
discontinuous only when θ =
discontinuous onl ...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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at Integral University, Lucknow, 06.06.2024
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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
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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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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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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2. Logarithmic and exponential forms
• Log 2 8
• Log 3 81
• Log 6 36
2 to the power of ?? gives 8
3 to the power of ?? gives 81
6 to the power of ?? gives 36
What is the base 2 logarithm of 8 ?
What is the base 3 logarithm of 81?
What is the base 6 logarithm of 36?
3. Logarithmic and exponential forms
• Log 2 8 = 3 23 = 8
• Log 3 81 = 4 34 = 81
• Log 6 36 = 2 62 = 36
2 to the power of 3 gives 8
3 to the power of 4 gives 81
6 to the power of 2 gives 36
What is the base 2 logarithm of 8 ? 3
What is the base 3 logarithm of 81? 4
What is the base 6 logarithm of 36? 2
25. • The point P on the curve 𝑦 = 9 𝑥 has y-coordinate equal to 150. Use logarithms to find the
x-coordinate of P, correct to 3 significant figures.
Using rules of Logs
26. • The point P on the curve 𝑦 = 9 𝑥 has y-coordinate equal to 150. Use logarithms to find the
x-coordinate of P, correct to 3 significant figures.
✓ 150 = 9 𝑥 Log 150 = Log 9 𝑥
Log 150 = 𝑥 Log 9
∴ 𝑥 =
Log 150
Log 9
= Log9150 = 2.28
Using rules of Logs
27. • Given that Log 𝑥 5𝑦 + 1 − Log 𝑥 3 = 4, express y in terms of x.
Using rules of Logs
28. • Given that Log 𝑥 5𝑦 + 1 − Log 𝑥 3 = 4, express y in terms of x.
✓ Log 𝑥
5𝑦+1
3
= 4
𝑥4
=
5𝑦 + 1
3
3𝑥4
= 5𝑦 + 1
∴ 𝑦 =
3𝑥4 − 1
5
Using rules of Logs
29. • Use logarithms to solve the following equation, giving the value of x correct to 3 s.f:
• 7 𝑥
= 2 𝑥+1
Using rules of Logs
30. • Use logarithms to solve the following equation, giving the value of x correct to 3 s.f:
• 7 𝑥 = 2 𝑥+1
✓ Log 7 𝑥 = Log 2 𝑥+1 → 𝑥 Log 7 = 𝑥 + 1 Log 2
𝑥
𝑥+1
=
Log 2
Log 7
= 0.356
𝑥 = 𝑥 + 1 0.356 = 0.356𝑥 + 0.356
𝑥 − 0.356𝑥 = 0.356
0.644 𝑥 = 0.356
∴ 𝑥 = 0.356 ÷ 0.644 = 0.553
Using rules of Logs