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!
