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Logarithms - What is e?

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Dec 08, 2016

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Logarithms - What is e?
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  • What is ‘e ’?
  • The two most common logarithms used
  • are log to the base 10
  • and log to the base e.
  • Why are they the most commonly used logarithms?
  • How do we really understand them?
  • Log to the base 10 is kind of intuitive.
  • It’s easier to talk in multiples of 10.
  • Ten, one hundred, one thousand and so on…
  • And log to the base ten gives us an easier scale to work with.
  • Log 10 to the base 10 is 1
  • log 100 to the base 10 is 2
  • log 1000 to the base 10 is 3…
  • and we can see that the multiples of 10
  • can be managed with a scale of natural numbers!
  • But Log to the base e is what I am deeply interested in!
  • It’s called the NATURAL log.
  • And is also written as LN.
  • Yes log to the base e is also written as LN.
  • So log 20 to the base e
  • can be written as LN of 20…
  • the natural log of 20.
  • What is this e?
  • Some call it a magical number,
  • some call it an irrational constant,
  • some call it the Euler’s number…
  • but the harsh truth is this:
  • Very few people actually understand what e is!
  • To get to e, we first need to understand GROWTH!
  • Let’s say a particular thing DOUBLES every time period.
  • On this time line, this is today, and every unit is one time period.
  • Assume you have a dollar with you!
  • At the END of the first time period, this dollar doubles, and you have 2 dollars!
  • At the end of the second period, 2 dollars double to become 4 dollars;
  • and at the end of the third period we have 8 dollars!
  • How do we look at this growth?
  • First, we see that the numbers at the end of the time periods are powers of 2.
  • It’s in the form 2 to the power x
  • where x is a non-negative integer.
  • 2 raised to 1
  • 2 raised to 2
  • 2 raised to 3
  • and so on.
  • This was one way in which we looked at Double Growth.
  • Another way to look at it is that there is a 100% growth every time period.
  • One plus ‘100% of one’ gives us 2.
  • One dollar was increased by 100 percent to get 2.
  • In the second time period, two dollars grow to four dollars.
  • Two plus ‘100% of 2’ gives us 4…
  • and so on.
  • So doubling the value is the same as a 100% increase!
  • So ‘2 raised to x’ can also be written as
  • ‘1 plus 100%’ raised to x
  • It’s like you are getting a ‘100% return’ on your investment.
  • But hold on… we are making an assumption here.
  • We are assuming that growth happens in a DISCONTINUOUS fashion.
  • We are seeing growth in steps here.
  • What about the time in BETWEEN two time periods.
  • We are seeing no growth in between.
  • No growth, and it suddenly doubles.
  • Again no growth, and suddenly doubles.
  • But hey, that’s not how nature functions!
  • Everything… or every kind of growth happens GRADUALLY.
  • If your height today is 4 feet,
  • you suddenly won’t be five feet a year later.
  • Your height gradually grows.
  • When we started, we used to get around 30 views a day.
  • And after a year, we started getting around 4000 views a day.
  • It doesn’t mean our view count just jumped one fine day.
  • It GRADUALLY increased!
  • So growth in nature is never really discrete or discontinuous.
  • Let’s see how it really works.
  • We take the example of a dollar growing over one year at a 100 percent growth rate.
  • First, we look at the annual growth!
  • Based on what we saw, at time zero, we would have one dollar.
  • And at the end of the year we will have 2 dollars.
  • This doesn’t seem right because all the interest cannot appear on the last day.
  • To make it slightly better, let’s divide the year into two equal parts.
  • 6 months, and 6 months.
  • Splitting that 100%, the growth would be 50% in the first year and 50 percent
  • in the second.
  • It would look like this.
  • Our initial dollar earned 50 percent interest in the first half to give us 50 cents more.
  • Now what happens in the second half?
  • ‘1 dollar 50 cents’ remains as is.
  • The growth is 50 percent.
  • So 50 percent of 1 dollar will be 50 cents.
  • And this time, the 50 cents also earn a 50 percent interest.
  • That will be 25 cents.
  • This 1.5 is the sum of our original dollar, and the 50 cents we made here.
  • So at the end of the first year,
  • we have our original dollar, then we have the dollar that
  • our original dollar made, AND we also have the 25 cents that these 50 cents made!
  • A total of ‘2 dollars 25 cents’.
  • This is better than doubling.
  • If we want to understand this using a formula,
  • it would be ‘one plus '100% over 2'
  • the whole squared.
  • We had half the growth rate over two time periods.
  • This is also referred to as SEMI ANNUAL growth.
  • Let’s push ourselves further!
  • What if we had FOUR equal time periods in a year?
  • We have divided one year into four quarters.
  • This is how it would look!
  • Looks messy, but is actually very simple if you’ve understood the concept!
  • Its 25 percent growth every quarter.
  • The formula would change to 1 plus ‘100% over 4’, the whole raised to 4
  • We would approximately get ‘two point four four one’ dollars at the end of the first year
  • I suggest you pause the video and understand the quarterly growth diagram really well.
  • This 100% is nothing but one.
  • If two time periods, then we have 2 here.
  • If 4 time periods, then 4 here.
  • So the formula for n time periods
  • would be 1 plus ‘1 over n’, the whole raised to n
  • Clearly, more the number of time periods,
  • higher will be the returns.
  • This will give us the dollar value in the end!
  • I probably know what your greedy brain is thinking.
  • Is it possible to get UNLIMITED money?
  • Let’s make a table now.
  • Number of time periods, and the dollar value in the end.
  • If it’s just 1 time period, the dollar value is 2.
  • If two time periods, then 2 dollars 25 cents.
  • If 4 times periods, then 2 dollars and 44 cents.
  • If I divide the year into 12 equal time periods,
  • my return will be higher than this!
  • If I divide it into 365 equal time periods, it will be even higher!
  • This tells us that a dollar at the start of the year will become these many dollars at
  • the end of one year, if the number of time periods is 365.
  • So if we increase this number significantly…
  • that is if we increase the number of time
  • periods significantly, will this number also increase significantly?
  • Ok here are a few more calculations.
  • The number of time periods here is one million!
  • Notice that the returns improve yes..
  • but they converge around a value which approximately
  • equals 2.718
  • And THAT is your beloved e.
  • We can’t get infinite money after all.
  • What would be a layman friendly explanation for e then?
  • It is the MAXIMUM possible result…
  • after continuously compounding…
  • a 100 percent growth…
  • over one time period!
  • Yes, that’s e.
  • Don’t forget, we had assumed a 100 percent growth here.
  • And that’s what e is.
  • It’s the maximum we get after a 100 percent continuous compounding growth, over one time period
  • Notice what compounding does!
  • The first result is 100 percent without compounding.
  • 1 dollar would become 2 dollars.
  • But after continuous compounding,
  • 1 dollar will be become 2.718 dollars
  • approximately.
  • That would be a growth rate of 171.8 percent.
  • That’s like the maximum growth we can have.
  • So e is approximately 2.718.
  • It’s an IRRATIONAL number…
  • which means the digits after the decimal point do not
  • repeat and go on forever!
  • Just one last question…
  • What if the growth rate and the time periods change?
  • Will e still help us?
  • Absolutely!
  • There’s no problem at all.
  • In general, the growth after continuous compounding
  • is given as e to the power ‘r times t’.
  • Where r is the rate and t is the number of time periods.
  • So if we have a 200 percent growth for 5 years,
  • then it would be defined as e to the power
  • ‘2 times 5’.
  • We squared e to include 200 percent growth,
  • and we raised it to 5 as there are 5 time periods!
  • It’ll give us e to the power 10.
  • e is nothing but the MAXIMUM possible result…
  • after continuously compounding a 100 percent growth
  • over 1 time period!

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What is e? What is the Natural Logarithm? Watch this video to know the answers. To view the entire course, visit https://dontmemorise.com/course/view.php?id=179

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