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let's say if b is equal to two then we see that okay the curves they don't touch and now the
quick question for you guys is that should we increase the b or should we reduce the b
the answer is we should reduce the b so let's try 1.5 okay they still don't touch but we can
see that they are getting closer so let's go ahead and try 1.4 okay they do touch but this time they
touch it at two points we don't want that i just want one point so now we can see that the b value
must be in between of 1.4 and 1.5 so let's just keep that in mind now how can we solve this though
technically this is just a typical one question whenever we have two curves touching each other
in a tension way then we will just have to do the following the first thing is that we'll just
make the functions equal to each other so we are looking at b to the x being equal to log base b of
x good and then and then in order for us to have they touch each other like this in a tangent way
we also have to make sure that they have the same derivative so let's look at this and differentiate
that and we will get b to the x and multiplied by l and b and if we differentiate this we get one
over x and then l and b cool so we do have a system of equations but how to solve this
shall we okay this is b to the x so should we just put this right there and um do it
and try if you put this right here you can solve for x in terms of b
and i will tell you this right here will give us x being equal to
e to the lambda w function of 1 over l and b i'm not kidding alright
and then you can go ahead and put this back to here and here and then try to solve for b
or you can put this back to here and here and try to solve for b but let me tell you
don't do this because it will be horrendous so what do we do then here is the deal let's take
a look at this picture again whenever we have an original touches in first
well at this point we can see what's so special about this point well let me give you guys a hint
how can we get the inverse from its original if we look at the picture well the answer is we take
a look at the original and we flip it about the line y is equal to x namely this diagonal line
so as you can see right here if they do intercept they will have to intercept each other on this
line so instead of looking at b to the x is equal to log base b of x in fact we can just say let's
go ahead and make b to the x being equal to just x or log base b of x just equal to x but in order
for this idea to work let me just write down this note for you guys we must have the function to be
increasing so that whenever we look at the equation f of x being equal to its inverse
by the way this is not a functional equation that we're solving i know it looks like a functional
equation but here we're just trying to find out an x value or maybe more to make this equation work
all right instead of solving this equation we can actually just solve f of x
being equal to x or if you would like and look at the inverse version of it as well
and either this or that will make the whole equation also so much easier and just a quick
shout out to mu prime map because i learned this from him and if you guys want to see a
detailed explanation go ahead and check out his video let's use b to the x to be
x and then we do the same thing we will have to make sure that they have the same derivative
so differentiate this we get b to the x times l m b and then differentiate that we get one
it's also so much easier isn't it even though this system equation is easier but still kind of tricky
though so be really careful what we can do first is that we notice b to the x is just equal to x
so we can of course put this x right here for that so we are going to get x times lmb being equal to
1. that's nice but now what well i want to produce an l and b and maybe let's use the
first equation again so have a look right here let me write down b to the x is equal to x again
but i'm going to take the natural log on both sides so now first we can put the x
to the front but you see we have the l and b so here we have x l and b this is done h
and that will just be equal to what l and x and guess what we know that x l and b
is equal to l and x so the whole thing here is actually just very nicely equal to l and x
very nice huh so ladies and gentlemen l and x is equal to 1 and of course just
exponentiate both sides we get x being equal to 1. no just kidding e x is equal to e
yeah all right but we're not done yet because this is just x we want to find
out what b is so of course we can put down the e to this x and not x so we are going to get
okay let me just put this down we have b to the e power being equal to e
and ladies and gentlemen to solve for b we can just raise both sides to the one over e power
like so so that this end will cancel and we will just get b being equal to e to the 1 over e power
or if you would like you can write this as the east root of e which is really really cool and
as we talked about the earlier this right here is indeed approximately one point four four
four six six seven eight and then six and so on so what do you guys think have a look