What is e and ln(x)? (Euler's Number and The Natural Logarithm)

exponential functions are some of the most important functions in mathematics and it can be used to model all sorts of different scenarios from measuring population growth and population change like for bacterium or a certain virus that's going around all the way to the financial sector such as interest from banks now let's focus on the financial side and talk about compound interest which is historically where the number e or all is constant was first discovered so say you rolled over to a band that gives you a hundred percent interest after one year if you give your money to them and what this means is after one year you get 100 of the money you put into the bank on top of the money you already have so then you end up with double the money you originally had now let's say you roll over to another bank but instead gives you 50 interest every six months now would it be better to put your money into this bank or the other bank well let's calculate it say you had p amount of some currency then for a bank with 100 interest at the end of the year we would end up with p plus p which can be factorized as p times one plus one which is just 2p now for a bank that gives 50 interest every six months we start with p and then six months down the line we get one half p or fifty percent p more and then six months after that we end up with 50 of the amount that we already have added on top now this amount can be simplified and factorized to p 1 plus a half all squared and this is equal to 2. 25 times p so you see we end up with more if we use the 50 twice rather than 100 interest once at the end of the year now what about 33 interest every one third of the year or every four months well we can do the same thing and we see that at the end of the year we end up with p times one plus a third all cubed which is equal to two point three seven times p now we see that this value is getting larger and larger the more times we have interest imposed now if we kept going then we see that 100 over n interest every one nth of the year gives us a final value of p times one plus one over n to the power of n and because it's getting larger and larger we want to find the maximum value that it can be to get the most amount of money that we could get so essentially we want to know the limit as n approaches infinity of one plus one over n all to the power of n and actually we can calculate this number and it ends up being this irrational number 2. 71828 dot dot dot and this number is famously written as the letter e similar to the way that we write 3.

141. as being the number pi and e is known as euler's constant and what this means is that the maximum amount that we could end up with is e times p or 2. 7 ish times p now we see that as n approaches infinity one over n approaches zero and n in the exponent approaches infinity of course but if we make the substitution k equal to one over n then k approaches zero can replace n approaching infinity and we get the limit as k approaches 0 of 1 plus k all to the power of 1 over k and this is another more useful way of writing the number e let's now try to differentiate exponential functions say we had y equal to 2 to the x then what is d y d x well we remember that d y d x is equal to the limit as h approaches zero of f of x plus h minus f of x all divided by h now if we take f of x equal to 2 to the power of x then we have the limit as h approaches 0 of 2 to the x plus h minus 2 to the x all divided by h now by an important rule of exponents we can write 2 to the x plus h as 2 to the x times 2 to the h then factorizing in the numerator we get 2 to the x times 2 to the h minus 1 all divided by h now we can interchange 2 to the x and the limit as 2 to the x has no dependence on h now this limit we can actually calculate and we get some irrational number 0.

6931 dot dot dot now this value also has a symbol which we'll see later but for now let's call this constant k2 so the y dx is equal to 2 to the power of x times some constant k2 now we can do the same for y equals 3 to the x and we get d y d x is equal to 3 to the x times some other constant k 3. and for 4 to the x we do the same thing and we get d y d x is equal to 4 to the x times some constant k 4. and we can keep going like this and we see that if y is equal to a to the x then d y d x is equal to a to the x times some constant k now this is great we found what d y d x is for exponential functions but each time we have to calculate this constant k from scratch so can we make k equal to one that is can we find a value a such that if y is equal to a to the x then d y d x is also equal to a to the x well we know that we can write d y d x as a limit and we can then divide both sides by a to the x and we see that this left side is actually our constant k from before so we want our value k to be equal to one so now what value of a should we choose well let's try a equal to e euler's constant and we remember that e is equal to the limit as h approaches zero of one plus h all to the power of one over h now we don't need to write the same limit twice and we see that h cancels in the exponent then the ones cancel in the numerator leaving the limit as h approaches zero of h over h and we know that h over h is just one and since there are no h's left we can remove the limit and we get one this means that if y is equal to e to the x then d y d x is also equal to e to the x and because the derivative is itself this makes y equal e to the x one of the fastest growing functions in mathematics and in fact it's scarily quick because this means the spread of infectious diseases is extremely fast and only gets faster now let's take a look at one of the slowest functions in mathematics and it's very related to one of the fastest functions in mathematics and that function is known as the natural logarithm now we know that 2 squared is equal to 4 and 2 cubed is equal to 8 but 2 to the power of what is equal to 6.

now we can write this number as log base 2 of 6 where log base 2 is a function and to see how we get this function let's consider y equal to log base 2 of x now we have a graph of y is equal to 2 to the x and if we reflect this graph in the line y equals x then we get the inverse function y equals log base 2 of x now let's replace 2 to the x with e to the x and remember e is a number 2. 71828 dot dot then it's reflection in y equals x becomes y equal to log base e of x now log base e of x is famously written as ln x which stands for logarithmis naturalis which is latin for natural logarithm now logarithms have some very important properties namely the logarithm of a product is the sum of logarithms which also implies that ln of x to the k is equal to k l and x and since exponentials and logarithms are inverse functions then e to the lnx is equal to x now with this we can also see that ln of x divided by y is equal to ln x minus lny now the natural logarithm is one of the most important functions in mathematics and we can start to see its usefulness when we differentiate it so take y equal to l and x and we know that d y d x is equal to the limit as h approaches zero of f of x plus h minus f of x all divided by h and substituting in we get ln of x plus h minus ln x in the numerator and now from one of the properties we can rewrite the numerator as ln of x plus h divided by x and let's write this divided by h to the side as one over h and we know from another property that we can bring this one over h to the exponent now we see here that x cancels which leaves us with 1 plus h over x and similarly to before let's make the substitution n equal to h over x so first we change the limit as h approaches zero to the limit as n approaches zero because when h approaches zero n also approaches zero and then we have one plus n in the brackets and one over nx in the exponent now we can write one over nx as one over n times one over x and by the one of the logarithmic properties again we can bring one over x to the front now we can swap the limit and natural logarithm and we see that this part here is our number e and since ln of e is equal to one we have that when y is equal to ln of x d y d x is equal to one over x and this unbelievable connection has very important consequences for example if any of you have done integration then you know that you cannot calculate the integral of one over x dx the normal way but here we show that this value is equal to ln of x now this relation also is the reason why the harmonic series which is the sum of all one over n diverges to infinity and a very useful trick is that we can write a to the power of x as e to the ln of a to the x which is just e to the x lna then by the chain rule we see that the derivative of a to the x is equal to a to the x times ln a and this lna is a constant k from before so instead of calculating k from the limit we now know what that constant should be and this is the reason why we always see exponential functions written as e to the kx rather than a to the power of x and as a real-life example we can measure the radioactive decay or half-life of a radioactive substance and what happens is over some period of time the mass of the substance is halved which gives a graph like y equals one half to the power of x now we can rewrite this as y equal to l another half x which is just e to the minus 0. 693.

x and now we can do whatever we want with it now definitely make sure you get used to these functions and know them inside and out as you will see them a lot throughout your mathematical journey so if you liked the video be sure to drop a like on it and hit that subscribe button if you want to learn explore and master more mathematics and head over to mathesy.