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okay in-studio lets talk about how can take you of a natural log of a negative number of course this way is not possible in the real world you have to go to a dark site for this let me show you how first of all we know that negative two is the same as negative 1 times 2 and when we have a product instead of a natural log we can break this into natural log the first one plus the natural log the second one so we get the first one as being natural log of negative 1 and then we add the second one is natural log of 2 so as you can see this right here if totally okay but this right yes totally now okay we have to fix this right here and to do so we are going to go to the complex plane for it so let me just write down our ear right here for the real axis and this right here Ill put up I am for the imaginary axis do not put on capital R do not put on capital C and like somebody back in 2017 dont do that anyway negative 1 on the complex plane is going to be right here because its just the same as a negative 1 plus 0 I so its on the real axis and now dont forget that when we have a complex number a + bi this right here is in the standard form which you can also call this to be the Kardashian form or the Cartesian form up to you this right here can change it to the polar form namely we can get our e I theta and to do so we need two things R is the distance from the

origin to the point so as you can see we move to left one time thats going to give us R equals 1 and the theta is you start from here you have to rotate well in this case we have to rotate 180 degrees but we are all those note we have to say PI theta okay see this PI and as you can see this number negative 1 we can write this as 1 for our x is the e and is 2i and theta is pi I guess this is the famous identity if you add one on both sides by anyway negative one is pretty much eetu the I pi and then we can put this back for the negative one instead of the natural log they cancel each other out very nicely so lets see here we get natural log of negative one was again use e to the I pi and then we are going to just add the natural log of two after that well as I said this right here cancel each other out and which is scared I PI isnt it yes yes of course right here lets just put them I pi and then of course in the end here we put down plus natural log of two so thats pretty much it but in fact this is only the first answer this is only the principal value in fact we have a lot more how many yes infinitely many why because this is not the only angle that can cares to here because of course you can rotate here again pi plus 2 pi of course dont give us 3 PI and can go to the again which dont give us 5 hi and so on so and so on and of

course you can also go backwards you can have the negative PI then then do you again which is going to keep you negative 3 PI and the negative 5 PI and so on so on so of course so the truth is once you have an angle PI we are going to plus 2 and pi means that you can keep rotating like this situation here so I will come here for this pi I will just go ahead and add to empower 2 pi on I think Ill just do like this 2 pi and point out where m is the integer so let me just indicate that I guess can comment down below YT I put em in stealth and anyway this is the angle so I should put on the Cs right this is the actual state here in general and then you multiply it by I and of course you can factor things out but let me just go there I will just add this right here and then from here as you can see we pretty much will have this record right here of course as you can see we have the pine come up so we can factor that out so oh no perhaps this is the punchline what were gonna do is this is equal to let me just write down the well I pi here at the end and then of course when you factor out the PI you have 1 + 2 m but which was pretty as 2m plus 1 this is representing a art multiple because M is a integer right so thats just some algebra and then in the end here you add L 2 like this so this is it and of course you have to indicate m is

an integer and Ill just box this right here for you guys so this is hope and in fact this is not the first time that we have seen that you to do this kind of things for example back in the days when you did not who you have enough where you had not learned about complex numbers yes square root of 2 is just cool – is no 1 point 4 and so on but if you have a negative inside of a square root of 2 like if you have a negative inside of a square root whats the ends of this well all you have to do this just make that into an I and multiplied by square root of 2 thats pretty much it and some people will argue either plus minus all that stuff but I will attack you I was just leave the principal square root of negative 1 I write here for you guys so the conclusion is when you have natural log of negative number so the natural log of that number stays the same but you change that to positive and you have to have this part oh no I can write this down for you guys as a home law so this right here is here as you can see if X is greater than 0 negative X will be negative Ln of negative x right here it will be this so thats pretty much it if you guys havent seen this kind of things before well hopefully like it however you enjoy this and if you do you should give me a like and also help to share the videos and if youre new to my channel be sure to subscribe thank us so much and as always thats it

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