Algebra 5.5 - Logarithms
Summary
TLDRIn this algebra series video, Matthew Sorbo explores the concept of logarithms through the lens of Warren Buffett's philanthropy. He explains how logarithms are the inverse of exponentiation, using examples to demonstrate their function. The video then connects logarithms to the idea that happiness increases logarithmically with income, showing diminishing returns as income rises. It discusses Buffett's decision to give away most of his wealth, suggesting that beyond a certain point, money doesn't buy happiness but giving can. The video concludes with a discussion on the U.S. tax system and Buffett's push for the wealthy to pay a fairer share of taxes.
Takeaways
- đ Warren Buffett, known as the 'Oracle of Omaha', has pledged to give away more than 99% of his wealth to philanthropy, emphasizing that additional wealth does not proportionally increase happiness or well-being.
- 𧟠Logarithms are mathematical functions that can model the relationship between wealth and happiness, showing that happiness increases logarithmically rather than linearly with income.
- đ° The concept of diminishing returns in happiness is illustrated by the logarithmic model, where a significant increase in income results in a smaller increase in happiness compared to lower income brackets.
- đ A logarithmic graph of income versus happiness shows a curve that levels off at higher incomes, indicating that additional wealth beyond a certain point has less impact on happiness.
- đ The initial increase in income from poverty to middle class provides a larger boost in happiness due to meeting basic needs, whereas moving from middle to high income has a smaller impact on happiness.
- đ Maslow's hierarchy of needs is referenced to explain why money's ability to buy happiness is limited once basic physiological and safety needs are met.
- đ” The United States has a marginal tax system where different income brackets are taxed at different rates, which aligns with the idea that money earned beyond a certain point contributes less to personal happiness.
- đŒ Warren Buffett has advocated for the 'Buffett Rule', which aims to ensure that the wealthy pay a higher percentage of their income in taxes to address the issue of tax loopholes and different taxation of investment income.
- đ€ The video script prompts a discussion on whether the wealthy should pay a higher percentage of their income in taxes, using the logarithmic relationship between income and happiness as a basis for the argument.
- đ The script encourages viewers to consider the impact of wealth on happiness and societal contributions, suggesting that giving away wealth can enhance one's sense of belonging and esteem more than personal consumption.
Q & A
What is the main topic discussed in the video script?
-The main topic discussed in the video script is the relationship between wealth and happiness, specifically using logarithms to model this relationship and exploring Warren Buffett's views on philanthropy.
Who is Warren Buffett and why is he mentioned in the script?
-Warren Buffett is often referred to as the 'Oracle of Omaha' and is one of the most successful investors in the United States. He is mentioned in the script to illustrate the concept of logarithms and to discuss his philanthropic approach to wealth.
What is the significance of logarithms in the context of this video script?
-Logarithms are used in the script to model the relationship between income and happiness. They help to demonstrate the concept of diminishing returns, where increases in income lead to smaller increases in happiness.
How does the video script use Warren Buffett's wealth to explain logarithms?
-The script uses Buffett's wealth to show that even a significant increase in wealth, such as a 100,000 dollar raise, would only result in a minor increase in happiness, illustrating the logarithmic nature of the happiness-income relationship.
What is the concept of diminishing returns as it relates to happiness and income?
-Diminishing returns in the context of happiness and income refers to the idea that as income increases, the additional happiness gained from each subsequent increase in income becomes smaller.
How does the video script use Maslow's hierarchy of needs to explain the relationship between money and happiness?
-The script uses Maslow's hierarchy of needs to show that money is more effective in increasing happiness when it is used to fulfill basic needs (lower rungs of the pyramid) rather than for luxuries or self-actualization (higher rungs).
What is the 'Buffett Rule' mentioned in the script, and what does it propose?
-The 'Buffett Rule' is a proposal by Warren Buffett to change tax codes to ensure that the wealthy pay a higher percentage of their income in taxes, not just a higher dollar amount, by eliminating loopholes and raising taxes on investments.
Why does the video script suggest that the wealthy should pay a higher percentage of their income in taxes?
-The script suggests that the wealthy should pay a higher percentage of their income in taxes because higher incomes do not increase happiness as much as lower incomes do, and the tax revenue can be used to address societal needs more effectively.
What is the marginal tax system, and how is it discussed in the script?
-The marginal tax system is a progressive tax structure where different portions of income are taxed at different rates. The script discusses it in the context of how it benefits individuals with lower incomes and generates more tax revenue from higher incomes.
How does the video script use the concept of logarithms to explain Warren Buffett's decision to give away most of his wealth?
-The script explains that because of the logarithmic relationship between income and happiness, Buffett's happiness would not significantly increase with additional wealth. Therefore, he chooses to give away most of his wealth to increase his sense of belonging and esteem by helping others.
Outlines
đ Introduction to Logarithms and Warren Buffett's Wealth
The video script introduces the concept of logarithms through the lens of financial success, exemplified by Warren Buffett, known as the 'Oracle of Omaha.' It discusses how investing with Buffett in 1964 could have led to significant returns by 2018. The script then poses the question of what Buffett plans to do with his wealth, hinting that he intends to give away most of it to philanthropy. The lesson transitions into explaining logarithms as the inverse operation of exponentiation, using examples to demonstrate how logarithms can isolate variables in equations. The video encourages viewers to follow along with downloadable notes and introduces the logarithmic model for happiness based on income, suggesting that beyond a certain point, additional wealth does not proportionally increase happiness.
đ Happiness and Income: The Diminishing Returns of Wealth
This section of the script delves into the relationship between income and happiness, using logarithmic models to illustrate the concept of diminishing returns. It explains how a small income can lead to minimal happiness, but as income increases, the rate of increase in happiness slows down significantly. The script uses a table and a graph to visualize this relationship, showing that while a jump from $10 to $50,000 annually results in a substantial increase in happiness, moving from $50,000 to $100,000 only marginally boosts happiness. The discussion references Maslow's hierarchy of needs to explain why basic necessities have a more significant impact on happiness than luxuries. The video also explores the idea that higher incomes provide less additional happiness, aligning with the concept of logarithmic compression of large numbers into a more manageable scale.
đŒ Warren Buffett's Philanthropy and Tax System Discussion
The final paragraph of the script returns to Warren Buffett's decision to give away most of his wealth, emphasizing how this aligns with the idea that additional income does not increase happiness for those already wealthy. It discusses Buffett's stance on the tax system, particularly the marginal tax rates in the United States, and how they affect individuals at different income levels. The script highlights Buffett's proposal for the 'Buffett Rule,' which aims to ensure that the wealthy pay a higher percentage of their income in taxes. The video concludes by posing a question to the viewers about whether they agree with Buffett's views on taxation and encourages them to consider the material presented in the lesson to form their opinions.
Mindmap
Keywords
đĄLogarithms
đĄWarren Buffett
đĄPhilanthropy
đĄDiminishing Returns
đĄMaslow's Hierarchy of Needs
đĄIncome
đĄHappiness
đĄMarginal Tax System
đĄBuffett Rule
đĄLoopholes
Highlights
Introduction to logarithms and their application in analyzing Warren Buffett's wealth and philanthropy.
Warren Buffett's net worth exceeded 100 billion dollars as of 2021, and his investment returns are exemplified.
Buffett's plan to give away most of his wealth to philanthropy, leaving little to his family.
Explanation of logarithms as the inverse operation to exponentiation.
Practical demonstration of solving for x using logarithms in the context of Buffett's wealth.
The concept of wealth and happiness being modeled by a logarithmic function.
Analysis of how happiness increases with income, but at a diminishing rate, illustrated with a logarithmic model.
The significance of the first income increase from poverty to middle income on happiness.
Maslow's hierarchy of needs as a framework to understand the relationship between wealth and happiness.
Graphing the logarithmic relationship between income and happiness to show diminishing returns at higher incomes.
Discussion on the impact of a large income raise on happiness for someone already earning a high salary.
Warren Buffett's perspective on how earning more money does not increase his happiness, but giving away wealth does.
Introduction to the United States' marginal tax system and its implications on income and happiness.
Buffett's proposal for the 'Buffett Rule' to ensure the wealthy pay a fairer share of taxes.
Engagement with the audience to discuss whether the wealthy should pay a higher percent of their income in taxes.
Conclusion and invitation for viewers to reflect on the content and its implications on personal finance and societal structures.
Transcripts
hello mathematicians my name is matthew
sorbo covering the algebra series on sku
the script
today we'll be discussing logarithms
specifically
can money by happiness without further
ado
let's skew it
[Music]
welcome in to lesson 5.5 of the skew the
script algebra series today we'll be
discussing
logarithms specifically you may have
heard of warren buffett often dubbed the
oracle of omaha
he's one of the most successful
investors ever in the united states
his net worth as of 2021 was in excess
of 100
billion dollars if you invested 19
with warren buffett in 1964 it will be
worth almost
300 000 in 2018 and even more
today so given that warren has
101.3 billion dollars at his disposal
what does he plan to do with his fortune
he could
do he could buy 5 000 luxury yachts
1600 tropical islands he could buy 10
000 private jets all those sound pretty
tempting
but what does he actually plan to do
give most of it
away warren has said that his pledge
is that more than 99 of his wealth goes
to philanthropy during his lifetime
or at his death his family won't get
any of that money so this might be a
little surprising to hear
but that's exactly today's key analysis
why would warren buffett give away
most of his wealth if you'd like to
follow along with today's lesson check
out the link below feel free to download
or print out the guided notes
and work along with the video to start
we're going to be reviewing logarithms
uh logs for short you can see our handy
dandy little
image there essentially uh we can think
of inverse operations to start these
help us
undo other operations in math for
example
if we have x plus 10 equals 25 how do we
undo this operation and solve for x
well we have the addition operation so
the inverse is the subtraction operation
we subtract 10 from both sides
cancels out on the left side we get x
equals 15.
how do we undo this operation 2x equals
28 to solve for x
well we have 2 times x which is
multiplication
the inverse is division we divide by
divide by 2 sorry to get
2's cancelling on the left side and x
equals 14.
how about this operation 4 to the power
of x
equals 64. the operation here is an
exponent
and the inverse is not immediately clear
however as makes sense because we're
talking about it this this section
the true inverse is the logarithm and in
this example
we see that log base 4 of 64 equals x
which is a little confusing and we will
get into it logs essentially undo the
exponent and isolate
x so a good way to remember this is the
loop
we start with a base which in this case
is four and you can see it's in the base
of the log it's it's colored green here
we go to the argument which in this case
is 64. it's the argument in our log
function so log base 4 of 64 and it
loops all the way back to the exponent x
which is what we end up isolating in the
end so
you can always evaluate a log in your
calculator this is log base 4 of 64.
in this case it comes out to 3. 3 equals
x
and again you can use it with any
standard calculator we can always check
by plugging 3 back into our original
equation
we get 4 to the power of 3 does indeed
equal 64.
4 to the power of 2 is 16 times 4 again
is 64.
um and you can see it here 4 times 4
times 4 is 64.
that is 16 times 4 again is 64.
so now we've kind of reviewed what
logarithms are let's look at logs of big
and small numbers returning to warren
buffett's previous quote
he said were we to use more than one
percent of my wealth on ourselves
in him and his family neither our
happiness nor our well-being
would be enhanced in contrast that
remaining 99
can have a huge effect on the health of
welfare and
others so a key part of this quote is
that neither
his happiness or his well-being without
his families would be enhanced
it would have a huge effect on the
health and welfare of others
mathematically we can express this as
wealth and happiness
is modeled or can be modeled with a
logarithm
so let's look at our handy-dandy table
where we have yearly income
happiness on a 0 to 10 scale and we have
a model y equals log base 10
of x x here equals yearly income that's
kind of our
explanatory variable y happiness on a
scale of 0 to 10
10 being the happiest in this case we're
going to use base 10 logs
and because it's a very standard format
base 10 logs are often written without
the 10
just written as log so here we have y
our happiness equals log of x
our yearly income let's try plugging in
some values so we plug in our yearly
income of just
ten dollars um we're going to get y
equals log 10
which comes out to one so not very happy
uh
one on a scale of zero to ten not very
happy but that makes sense because
yearly income is quite small how about
if we plug in a way bigger number than
ten dollars
fifty thousand dollars a year as your
yearly income we get y equals log
fifty thousand that comes out to four
point seven so
much much happier than with 10 dollars
4.7 far higher than than one on the
scale of happiness
if we plug in a hundred thousand dollars
for our yearly income we get
a happiness of five and uh we can
continue in this way because we have log
y equals log of x but first we can look
and see that
when we go from ten dollars to fifty
thousand dollars in our income
we get a huge increase fifty thousand
dollar increase um and when we go from
fifty thousand dollars to a hundred
thousand dollars we also get a fifty
thousand dollar increase in income
um however when we go from ten dollars
to fifty thousand dollars
we get a big jump in happiness three
point seven points but from fifty
thousand to a hundred thousand
get a much smaller jump just point three
jump in happiness from those two points
so what's going on we had a fifty
thousand dollar increase in both but a
much bigger jump in happiness
we went from ten dollars to fifty
thousand dollars um
what's important to know is jumping from
ten dollars to fifty thousand dollars
this is jumping from
the poverty line to middle income what
does middle income give you
guaranteed shelter likely you have
health insurance from your job
your next meal is not an issue and you
have money for personal interest beyond
your uh disposable income you have
disposable income for your
personal interest beyond your basic
costs and necessities
um you can see there's a huge jump in
predictive happiness because
it's a big jump in quality of life as we
kind of detailed on the previous slide
when you jump from middle income to
fairly high income so fifty thousand to
a hundred thousand
you do get some benefits you have a
better house a nicer car
spending on luxuries but luxuries don't
increase happiness as much as
the guaranteed necessities of jumping
from poverty to middle
income um one way to encapsulate this is
with maslow's hierarchy of needs but you
can see here and this kind of outlines
the question of if money can buy
happiness
the higher you move up on this pyramid
the happier you
are the lower rungs of safety needs so
personal security employment resources
and physiological needs so the very
simple
air water food shelter you can't move up
on the rung until you've achieved the
lower
lower levels so you need to be safe you
need to have your physiological needs
cared for you cannot reach esteem
self-actualization level belonging if
you're st the higher rungs no ladder if
you're still worried about
food water and shelter on the lower
rungs money can help a lot with the
lower runs you can buy houses you can
buy food all that stuff
but not so much with the higher realms
it doesn't help as much with esteem
self-actualization love and belonging
so if we return to our table and we
continue plugging in higher values of
income
we see the trend continues another 50
000 increase
to 150 000 income per year just
increases happiness by 0.2
and if we add go up to 200 000 a year
happiness just increases by 0.1 to
5.3 um so again looking at this uh
from the top logarithms compress the
numbers right so we have these really
big numbers
logarithms compress it down to a smaller
scale they take big numbers which is the
early income
and they compress them more than the
small numbers so yearly income of 10
gets compressed to one
the yearly income of 200 000 gets
compressed all the way
to 5.3 that's a good property of
logarithms
this is good for diminishing returns
which in this case we'll talk about it
more
means big money a large salary earns
little extra happiness as compared to
early gains on on happiness so let's
actually look at graphing logs to drive
home
this intuition we take our table and
we'll work to visualize it
we plot yearly income on the x-axis and
happiness on the 0-10 scale on the y
axis
and we can just put in our dots from the
table here and
sort of think about what shape you see
we can draw this line here this
kind of curved line which represents y
equals log x
and remember our maslow's hierarchy of
needs pyramid
that first fifty thousand dollars in
income makes happiness increase
rapidly you get a big increase in
happiness because you can use it for
basic needs
air water food shelter security
all that sort of stuff but at higher
incomes we get diminishing returns
your happiness increases less rapidly
from the money you make because the
higher incomes don't help as much with
self-actualization
esteem and love and belonging so let's
think about a specific question
you work a very high income job you make
four hundred thousand dollars per year
let's imagine that you work super super
hard and you get a massive
one hundred thousand dollar raise that's
a big rate it's 25 of the money
that you make how much is your happiness
predicted to increase
so you go from 400 000 to 500 000
at la at y equals log of 400 000 your
happiness is 5.6
how about at uh log of 500 000 what does
your happiness go to
5.7 so you go from 5.6 to 5.7
just a 0.1 difference in happiness from
that
huge bonus and again this confirms the
fact that at higher incomes we see
diminishing returns
happiness increases much less rapidly at
higher incomes this kind of nails on the
point
money can't buy love and belonging can't
buy self-actualization
or necessarily esteem warren buffett
course
of course is way way way to the right on
this chart it extends a long way until
we see warren buffett
and now we can return to his quote where
he said that neither happiness or his
well-being
would be enhanced by this extra money
so if we think about how warren's way
over there on the scale of dimension
returns
if he can't get happier by earning more
money how can he
increase his happiness well he notes
that giving away 99
of his wealth has a huge effect on the
health and welfare of others
and as you can see by warren giving to
his community giving wealth to charity
and still saving enough to meet the
needs of himself
it increases his belonging and esteem
because he's helping
others and serving others in the
community so it helps him with
his top of the pyramid now that we've
kind of explored
warren buffett's decision making in his
uh personal finances let's turn to the
discussion
um specifically we'll be talking about
the united states which has a marginal
tax system
for example in 2021 if you earn your
first ten thousand dollars earned is
taxed at ten percent
your next thirty thousand dollars is
taxed at twelve percent
next forty five thousand dollars is
taxed at twenty two percent
etc and then earnings after five hundred
thousand dollars is taxed at thirty
seven percent
there are a couple of benefits of this
marginal system for example
for individuals that are not making as
much money their first
ten thousand dollars is has a lower tax
that's necessity money it's required for
safety needs
physiological needs that sort of thing
um whereas more tax revenue is actually
from luxury money so money earned past
250 000
which doesn't increase happiness as much
as we've seen um
or may not increase happiness as much as
we've seen uh we get
more tax revenue from that there's also
no back penalty for earning more so if
you earn
more than ten thousand dollars it
doesn't affect the taxes on the first
ten thousand dollars
that means it is never better off to be
working less which is
an important point to note there is one
catch
wealthy individuals hire very good
accountants who then find loopholes in
the system to lower their taxes
you also have income from investments
like warren buffett's stocks and bonds
those are taxed differently than normal
income so buffett
uh tried to establish the buffett rule
he wrote new york times op-ed
which was titled stop coddling the super
rich and it's linked here
um he essentially proposed this rule to
change tax codes to guarantee that the
wealthy pay a higher dollar amount and a
higher
percent of their income in taxes not
just a higher amount but a higher
percent
to do this you have to eliminate
loopholes and raise taxes on
investments which many of them have so
to turn to the discussion
do you agree with buffett that the
wealthy should pay not only a higher
dollar amount in taxes
but also a higher percent of their
overall income why or why not
and be sure to to use the material from
this lesson to support your answer
that's all for today thanks for joining
and we'll see you next time on skew the
[Music]
script
[Music]
Voir Plus de Vidéos Connexes
Konsep Dasar dan Sifat-sifat Logaritma Matematika Peminatan Kelas 10
LOGARITMA ITU GAMPANG!! Bahas Logaritma Kelas 10 | Study With Jerome Polin
How the rich avoid paying taxes
Does Money Actually Buy Happiness? A Psychiatrist Explains
01 - Simplify Rational Exponents (Fractional Exponents, Powers & Radicals) - Part 1
Billionaires Pay Lower Tax Than WORKING CLASS
5.0 / 5 (0 votes)