# Logarithms - What is e?

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Dec 08, 2016

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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.
• 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.
• 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!

### Description

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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