hello let's do some math for fun here we are going to find a solution to the equation e^x=ln(x)
and you can expect that the answer to this right here is not real
because if you graph y=e^x you will see that e^x is like this and ln(x) is
like that they will NOT cross each other but one good thing is that you know they are the inverse
of each other right anyway though the challenge is that can we find a complex solution to this and as
always please pause the video and think about this first and today we have this guy okay all right so
this right here is actually inspired by one of the recent video that I did on new programs channel
so guess go ahead and check that out where I say that if you have the function that's equal to its
inverse you can you actually just set out to be equal to X and solve it unfortunately that does
not give other solutions every single time however it will keep the other structure every single time
if the functions increasing and always increasing cap but the worst part here is that they don't
cross each other so that matter it's still going to work but I'm not going to go in there all right
anyway though hopefully you guys have a chance to try it now here we go let's see how we can do
this so start it with e to the X that's equal to Ln X yeah perfect then I'm going to just look at
e to both sides so that this and that work and so perfect and you will see that we can write
this s X has to be equal to what well e to the e to the X isn't it great so that's what exists and
this reminds me why I just say earlier if you put this is equal to X by the way that I would do this
approach right because actually I did the similar approach in one of my video which is gradually
older so you guys go ahead and check that out you were like all right here we go X is equal to that
huh so have a look X is equal to e to the e to the X right but if X just equal to e to the X
e to the X well if I write that down again I will put down e to the e and this is e to the e to the
X because of this so see X is equal to e to the e to the e to the e to the X but hold on because if
you were like you can look at this X right here as the blue X which is still up e to the e to the
X and of course I can put this down right here so you will see that this right here becomes e
to the e and then we have two more e in red and then we have two more e in blue but you know I'm
going to put on the X right here but you know this procedure can keep on going forever so you
know you can actually just put on dot dot dot so in other words X becomes what just e to the e to
the e to the e to the infinitely many E's Wow Wow the truth is that this right here does not count
search all right so you have to pay attention to if you can purchase for now anyway though we are
not doing real mathematics we are doing complex mathematics so we can go to the complex world
so have a look well what's eat would eat e to the older stuff hmm you know e to the e to the
e tattered are forever this right here it's equal to X but if you pay attention to just this portion
this is still each of the e to e infinitely many times so this right here is still equal to X so
in another word we can look at this as e to the X that's equal to X like this yeah good and now
how are you gonna solve it take you the natural log of both sides no otherwise know just know we
actually have to use the Lambert W function and this is how we can do it let me get by this on
both sides or justify this by e to the X like so and you will see the left-hand side is 1 and the
right-hand side becomes e X and I'll write this as e to the negative X power and the purpose for
me to do that is here we have negative x the pace right here e this is X which is almost the same I
just need to have negative it's ok let me just multiply the negative on both sides and then we
can apply the Lambert tap wave function so let me do it here and let me also do here and when we do
that the left hand side will just give us negative x right because the limbered up your function of
this and that I will just end up with this and the right hand side I'll put this right here which
is just a Lambert of you function of negative 1 and then I will just divide the x on both sides
you will see I mean not certify the example set divide the negative on both sides you will see
X is equal to negative W of negative 1 I this a looks like we only have one answer but the truth
is no because you actually can go to different branches of the Lambert W function if you pick 0
right here for the principal branch you end up with approximately against the complex number
so I'll just write this down you end up 0.318 yeah that's the real part and then you - 1.337 I
anyway hopefully gets on like this video I'll see
you guys next time and it's always that it's in it you know in my back