today we are going to figure this out i^x=2. how's this possible well last time I showed
you guys 1^x=2 already so of course that's possible right okay before I show you guys
this I would like to recall a question that we have done before how do we figure out I to the I
notice that the base and also the power they are both imaginary but the results are always real
to figure this out we first look at the base ID right here and put into the polar
form so we take a look at the real axis and also the Imagine axis I is right here
put this into the polar forming stat we have to find REI Theta R is the distance from here to
here it's just one unit up and then e is just e i is just I Theta is the angle from here to
here and please don't say 90 degrees because we are adults now so say pi over 2. so we can put I
right here as 1 for R and then e and then another I and then Theta which is pi over 2. but the truth
is this is not the only angle that will make this work because you can keep rotate this way or the
other way so technically we can add to um pi so this is how we end up with infinitely many answers
and then raise the whole thing to this ice power so I'll put this to the eye
then what we can do next is just multiply this and not together I times I is negative one so we see
this right here just give us e to the negative and then we have pi over 2 plus 2 um pi and of course
right here I actually denote that n is an integer so that's why we have infinitely many answers
and when n is equal to zero meaning that we don't have this part we just get e to
the negative pi over 2 and that's kind of like the first answer and that's called the principle
value so let me just write that down right here for you guys real quick the principle value
of I to the I is e to the negative pi over 2. last time when I showed you guys 1 to the X power is
equal to 2 of course the X cannot be real because otherwise 1 to a real power is always equal to one
and the problem is when we have one to a now real power we have infinitely many results
as well and the X formula I got for you guys last time one to that x value we have
infinitely many answers but one of them is two so you might be thinking do you really count
that as the solution some people say yeah some people may say no so I will leave that to you
to reduce confusion and the complexity of this video I just want to focus on the
principle for just the first answer and we do have nice results all right so how can we make
this happen same idea right first we have I to the X power equals two so of course we can rewrite the
i s this but because we just want to focus on the principle value we don't want to look at
that part so just write it as e to the I pi over 2. so right here we will have e to the I pi over
2 and then raise that to the X power and that's equal to 2. all right now this time start we have
e to the I pi over 2 x this is equal to two then we can just take the natural log on both sides
and choose the principle for you so you don't need to enter whatever other stuff right just this all
right so we have I pi over 2x it's equal to ln2 now I'm going to just multiply the reciprocal of
pi over 2 so we have 2 over Pi so you get x equals 2 and then we have the ln2 and the over pi and of
course we also divide the I right so I will also put a I right here but you know it I don't like
to be on the bottom I like to be on the top so let's multiply the top and bottom but I like this
together this right here will give us negative one so on all we get negative and let's write
the two first so 2i and then L and two over Pi so that's what we have that's pretty good
however even until I said this is the principle for you I mean just the principle for you of
the complex expression right I to this power that will be it however this is not the only solution
and that's the mistake that I had last time for the 1 to the X power you guys see it see it yes no
ready so nothing wrong with all this but this is just not enough check this out I will have to say
notice we know that I to the first is equal to i i squared by definition is negative one I to
the third power is equal to negative i i to the fourth power is equal to one and then it
repeats here I to the fourth power is equal to one so imagine if we have multiple four
that should also contribute to the X because that would just give us one anyway so in fact
the answer for this question is X is not just that I will put down minus 2 i l and 2 over Pi but we
will also need the multiple four because of that so I will just put down the four and at the front
this right here is the answer check this out I will check the answer for you so we have
the base that's I and then we have the 4N so this is the real number perhaps I also denoted and is
the integer here it's just a real number and then we have the minus oh I should also just write it
down red so minus 2 i l and two over Pi if we work this out do we end up with two of course it does
so what we do first is we can write this s i to the 4N and then times I to the negative 2 i
l and 2 over pi this is always equal to one i to the multiple four it's always equal to one so we
just have a nice number one right here and then to work this out this I let's just write the s e
to the I pi over 2 again so e to the I pi over 2 and then we will raise that to this power negative
2 i l n 2 over Pi ready for the cancellations Pi Pi cancel 2 2 cancel I times negative I will
give us positive one and then e to the L and two is just two so on o we do end up with a
nice number two so it does check [Music] then you will see that this is the answer that they gave us
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