Logarithms - What is e? | Euler's Number Explained 

                                                                               

What is ‘e ’? the 2 commonest logarithms used are log to the bottom 10 and log to the bottom e. Why are they the foremost commonly used logarithms? How can we really understand them? Log to the bottom 10 is quite intuitive. It’s easier to speak in multiples of 10. Ten, 100 , one thousand then on… And log to the bottom ten gives us a neater scale to figure with. Log 10 to the bottom 10 is 1 log 100 to the bottom 10 is 2 log 1000 to the bottom 10 is 3… and that we can see that the multiples of 10 are often managed with a scale of natural numbers! But Log to the bottom e is what i'm deeply interested in! It’s called the NATURAL log. And is additionally written as LN. Yes log to the bottom e is additionally written as LN. So log 20 to the bottom e are often written as LN of 20… the natural log of 20. what's this e? Some call it a magical number, some call it an irrational constant, some call it the Euler’s number… but the tough truth is this: only a few people actually understand what e is! to urge to e, we first got to understand GROWTH! Let’s say a specific thing DOUBLES whenever period. On this point line, this is often today, and each unit is just one occasion period. Assume you've got a dollar with you! At the top of the primary period of time , this dollar doubles, and you've got 2 dollars! At the top of the playing period , 2 dollars double to become 4 dollars; and at the top of the playing period we've 8 dollars! How can we check out this growth? First, we see that the numbers at the top of the time periods are powers of two . It’s within the form 2 to the facility x where x may be a non-negative integer. 2 raised to 1 2 raised to 2 2 raised to three then on. This was a method during which we checked out Double Growth. differently to seem at it's that there's a 100% growth whenever period. One plus ‘100% of one’ gives us 2. One dollar was increased by one hundred pc to urge 2. within the second period of time , two dollars grow to four dollars. Two plus ‘100% of 2’ gives us 4… then on. So doubling the worth is that the same as a 100% increase! So ‘2 raised to x’ also can be written as ‘1 plus 100%’ raised to x It’s such as you are becoming a ‘100% return’ on your investment. But hold on… we are making an assumption here. We are assuming that growth happens during 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 all kinds 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. once we started, we wont to get around 30 views each day . And after a year, we started getting around 4000 views each day . It doesn’t mean our view count just jumped one fine day. It GRADUALLY increased! So growth in nature isn't really discrete or discontinuous. Let’s see how it really works. We take the instance of a dollar growing over one year at a one hundred pc rate of growth . First, we glance at the annual growth! supported what we saw, at time zero, we might have one dollar. And at the top of the year we'll have 2 dollars. This doesn’t seem right because all the interest cannot appear on the Judgment Day . to form it slightly better, let’s divide the year into two equal parts. 6 months, and 6 months. Splitting that 100%, the expansion would be 50% within the first year and 50 percent within the second. it might appear as if this. Our initial dollar earned 50 percent interest within the half to offer us 50 cents more. Now what happens within the second half? ‘1 dollar 50 cents’ remains as is. the expansion is 50 percent. So 50 percent of 1 dollar are going to be 50 cents. And this point , the 50 cents also earn a 50 percent interest. which will be 25 cents. This 1.5 is that the sum of our original dollar, and therefore the 50 cents we made here. So at the top of the primary year, we've our original dollar, then we've the dollar that our original dollar made, and that we even have the 25 cents that these 50 cents made! a complete of ‘2 dollars 25 cents’. this is often better than doubling. If we would like to know this employing a formula, it might be ‘one plus '100% over 2' the entire squared. We had half the expansion rate over two time periods. this is often also mentioned as SEMI ANNUAL growth. Let’s push ourselves further! What if we had FOUR equal time periods during a year? we've divided one year into four quarters. this is often how it might look! Looks messy, but is really very simple if you’ve understood the concept! Its 25 percent growth quarterly . The formula would change to 1 plus ‘100% over 4’, the entire raised to 4 we might approximately get ‘two point four four one’ dollars at the top of the primary year I suggest you pause the video and understand the quarterly growth diagram rather well . This 100% is nothing but one. If two time periods, then we've 2 here. If 4 time periods, then 4 here. therefore the formula for n time periods would be 1 plus ‘1 over n’, the entire raised to n Clearly, more the amount of your time periods, higher are going to be the returns. this may give us the dollar value within the end! I probably know what your greedy brain is thinking. Is it possible to urge UNLIMITED money? Let’s make a table now. Number of your time periods, and therefore the dollar value within the end. If it’s just 1 period of time , 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 are going to be above this! If I divide it into 365 equal time periods, it'll be even higher! This tells us that a dollar at the beginning of the year will become these many dollars at the top of 1 year, if the amount of your time periods is 365. So if we increase this number significantly… that's if we increase the amount of your time periods significantly, will this number also increase significantly? Ok here are a couple of more calculations. the amount of your time periods here is one million! Notice that the returns improve yes.. but they converge around a worth which approximately equals 2.718 which is the one that you love e. We can’t get infinite money in any case . What would be a layman friendly explanation for e then? it's the utmost possible result… after continuously compounding… a one hundred pc growth… over just one occasion period! Yes, that’s e. Don’t forget, we had assumed a one hundred pc growth here. And that’s what e is. It’s the utmost we get after a one hundred pc continuous compounding growth, over just one occasion period Notice what compounding does! the primary result's one hundred pc without compounding. 1 dollar would become 2 dollars. But after continuous compounding, 1 dollar are going to be become 2.718 dollars approximately. that might be a rate of growth of 171.8 percent. That’s just like the maximum growth we will have. So e is approximately 2.718. It’s an IRRATIONAL number… which suggests the digits after the percentage point don't repeat and continue forever! only one last question… What if the expansion rate and therefore the time periods change? Will e still help us? Absolutely! There’s no problem in the least . generally , the expansion after continuous compounding is given as e to the facility ‘r times t’.the whole raised to n Clearly, more the amount of your time periods, higher are going to be the returns. this may give us the dollar value within the end! I probably know what your greedy brain is thinking. Is it possible to urge UNLIMITED money? Let’s make a table now. Number of your time periods, and therefore the dollar value within the end. If it’s just 1 period of time , 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 are going to be above this! If I divide it into 365 equal time periods, it'll be even higher! This tells us that a dollar at the beginning of the year will become these many dollars at the top of 1 year, if the amount of your time periods is 365. So if we increase this number significantly… that's if we increase the amount of your time periods significantly, will this number also increase significantly? Ok here are a couple of more calculations. the amount of your time periods here is one million! Notice that the returns improve yes.. but they converge around a worth which approximately equals 2.718 which is the one that you love e. We can’t get infinite money in any case . What would be a layman friendly explanation for e then? it's the utmost possible result… after continuously compounding… a one hundred pc growth… over just one occasion period! Yes, that’s e. Don’t forget, we had assumed a one hundred pc growth here. And that’s what e is. It’s the utmost we get after a one hundred pc continuous compounding growth, over just one occasion period Notice what compounding does! the primary result's one hundred pc without compounding. 1 dollar would become 2 dollars. But after continuous compounding, 1 dollar are going to be become 2.718 dollars approximately. that might be a rate of growth of 171.8 percent. That’s just like the maximum growth we will have. So e is approximately 2.718. It’s an IRRATIONAL number… which suggests the digits after the percentage point don't repeat and continue forever! only one last question… What if the expansion rate and therefore the time periods change? Will e still help us? Absolutely! There’s no problem in the least . generally , the expansion after continuous compounding is given as e to the facility ‘r times t’. Where r is rate and t is number of your time periods. So if we've a 200 percent growth for five years, then it might be defined as e to the facility ‘2 times 5’. We squared e to incorporate 200 percent growth, and that we raised it to five as there are 5 time periods! It’ll give us e to the facility 10. e is nothing but the utmost possible result… after continuously compounding a one hundred pc growth over 1 time period!

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