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 can be used to model all kinds of different scenarios, from measuring population growth and change, such as a bacteria or a certain virus circulating, to the financial sector. , like interest. from the banks now let's focus on the financial aspect and talk about compound interest, which is where historically the

number

e was first discovered or everything is constant, so let's say you move to a band that gives you one hundred percent interest after a year if you give your money to them and

what

this means is that after a year you get 100 of the money you put in the bank in addition to the money you already have, so you end up with double the money you originally had, now let's say that You move to another bank but instead it gives you 50 interest every six months, would it be better to put your money in this bank or in another bank?

Well, let's calculate it, let's say you have p amount of some currency and then for a bank with 100 interest at the end of the year. year we would end up with p plus p which can be factored as p multiplied by 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 later we get half p or fifty percent p plus and then six months later we end up with 50 of the amount we have already added above now this amount can be simplified and factored to p 1 plus half everything squared and this is equal to 2.25 times p So you see, we end up with more if we use interest 50 twice instead of 100 once at the end of the year.

Now,

what

happens with the interest 33 every third of the year or every four months? Well, we can do the same and see. that at the end of the year we end up with p multiplied by one plus one third all cubed, which is equal to two point three seven multiplied by p now we see that this value becomes increasingly greater the more times interest is imposed on us if If we continue, we see that 100 over n interest every nth of the year gives us a final value of p multiplied by one plus one over n to the power of n and because it gets bigger and bigger, we want to find the maximum value that can be to get the most amount of money we can get, so essentially we want to know the limit as n approaches the infinity of one plus one over n, all raised to the power of n, and we can actually calculate this

number

and it ends up being this.

Irrational number 2.71828 dot dot dot and this number is written as the letter e, similar to the way we write 3.141. since the number pi and e is known as Euler's constant and what this means is that the maximum quantity we could end up with is e multiplied by p or 2.7 times p. Now we see that as n approaches infinity, one over n approaches zero and n in the exponent tends to infinity, of course, but if we make the substitution k equal to one over n, then k approaches zero can replace a n as it approaches infinity and we get the limit as k approaches 0 of 1 plus k, all raised to 1 over k. and this is another more useful way of writing the number e, now let's try to differentiate exponential functions let's say we have y equals 2 to the power of x, so what is d and d x?

Well, let's remember that d and 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 from 2 to the power of x plus h minus 2 to the power of x all divided by h now, using an important rule of exponents, we can write 2 to the power of x plus h as 2 to the power of to h minus 1, all divided by h now we can exchange 2 with x and the limit as 2 with for now let's call this constant k2 so that y dx is equal to 2 to the power of x multiplied by a constant k2.

Now we can do the same for y equals 3 to the x and we get d and d x equals 3 to the x. some other constant k 3. and for 4 to the power of x we ​​do the same and we get d and d x is equal to 4 to the power of constant k now this is cool, we find what d and d is equal to a x then d and d x is also equal to a x? Well, we know that we can write d and d x as the limit and then we can divide both sides by a x. x and we see that this left side is actually our constant k from before, so we want our k value to be equal to one, so now what value of a should we choose?

Let's try with a equal to e Euler's constant and 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 units 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 left we can remove the limit and we get one, this means that if y is equals e to the power of x, then d and d x is also equal to e to x and because the derivative is itself, this makes y equal to e to It means that the spread of infectious diseases is extremely rapid and is only getting faster.

Now let's take a look at one of the slowest functions in mathematics and it is closely related to one of the fastest functions in mathematics and that function is known as

natural

logarithm

. Now we know that 2 squared is equal to 4 and 2 cubed is equal to 8 but 2 to the power is equal to 6. We can now write this number as log base 2 of 6 where log base 2 is a function and to see how we obtain this function let us consider y equal to log in base 2 of x now we have a graph of y is equal to 2 to the power of x and if we reflect this graph in the line y is equal to x then we obtain the inverse function y is equal to log in base 2 of x now let's replace 2 to the power of x with e to the power of x and remember that e is a number 2.71828 dot dot, so it is reflected in y is equal to famously as ln x, meaning

logarithm

natural

is, which now means natural logarithm in Latin.

Logarithms have some very important properties, namely, the logarithm of a product is the sum of logarithms, which also implies that ln from x to k is equal to k l and x and since exponentials and logarithms are inverse functions, then e to lnx is equal to x. now with this we can also see that ln of x divided by y is equal to ln that d and d x is equal to the limit when h tends to zero of f of x plus h minus f of x all divided by h and substituting we obtain 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 next to it as one over h and we know from another property that we can take this over h to the exponent now we see here that similar 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 as h approaches zero n also approaches zero and then we have one more n in parentheses and one over nx in the exponent now we can write one over nx as one over n multiplied by one over x and by one of the logarithmic properties again we can bring one over x to the front now we can swap the limit and the 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 and d x is equal to one over x and this incredible connection has very important consequences, for example, if any of you have done an integration , then you know that you can't compute the integral of one over x dx in the normal way, but here we show that this value is equal to ln of x now.

This relationship is also the reason why the harmonic series that 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 x, which is just e to the power of x lna, then according to the chain rule, we see that the derivative of a to the power of x is equal to a to the power of x multiplied by 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 why we always see exponential functions written as e to the kx instead of a to the x and as a real life example we can measure radioactive decay or the half-life of a radioactive substance and what happens is that over a period of time the mass of the substance is reduced by half which gives a graph like y equals half to the power of x now we can rewrite this as y equals l other half x which is simply e to at least 0.693.x and now we can do whatever we want with it.

Definitely make sure you get used to these features and know them inside and out, as you'll be seeing them a lot throughout your math journey, so if you liked the video, be sure to like it and hit the subscribe button if you want to learn. , explore and master more mathematics and head over to mathesy.com where you can find all my notes, my videos and if you want to support the channel, I hope you have a good morning and I'll see you all next time.