okay let's do some math for fun today I have a log triangle for you guys
what do I mean well let me show you first let me give you guys a right triangle and of course
because we're talking about a logarithm function let's talk about the natural log let me label
this as ln(x) and let's say this right here it's a little bit longer so let's say I'll end up ln(2x)
and then just for fun let's make this ln(3x) and the question is is it possible for us to find
an x value so that this is actually a legitimate right triangle of course the answer is yes because
otherwise how can we continue right and the truth is it seems like when you have a rectangle just
put on some like functions like want in a good pattern sometimes you can actually come up with
some really nice questions but anyway though as always please pause the video and try this first
done all right of course we will first have to use the Pythagorean theorem because we're talking
about the size of a right triangle so let's make this right here square and then we add this side
Square which is Ln of 2x square that and then make this equal to that square which is Ln of 3x
all right now here's a small problem because the input here is X and then the increase is 2x hence
3x they are different um you cannot do too much but don't worry because this is Ln of 2 times x
thanks to the log property we can break it apart yes ln2 plus L and X likewise we can do the same
thing right here so let me just keep it as how it is we have lnx square and then for this one
we have Ln 2 plus Ln X and then square and this is equal to ln3 Plus lnx Square now much better and
let's just go ahead and multiply our everything and hope for the best this right here stays so
I'm just going to keep it as Ln x square and for this of course we Square the first thing which
is going to be just our next one two square and then we add 2 times this and that so we have 2 L
and two and then L and x and lastly we have to add this Square so we have to put this down right here
and then I'm going to put on right hand side right here this is going to be equal to this Square
which is ln3 square and then plus 2 times this and that so we have 2 L and three and then Ln X and
lastly this guy squared which is L and X squared pretty cool all right now we can just continue and
then just finish this it looks pretty nice because now we have a quadratic equation in terms of L and
X all right so no matter just cleans things up and here's the good thing we see that on
the 11th side we have this L and x square likewise the right hand side in Falcon cancel this and that
out but we still have this Ln x squared right let's just deal with it l and x and then Square
okay let me put this and that on one side and then of course factor out the lnx
so keep this right here that will be plus 2 ln2 and then this to the other side so it becomes a
minus 2 L and three and then the lnx right here and lastly I'll put this to the other
side as well so we have plus L and two Square minus ln3 square and this is equal to zero
cool now let's see if there's anything that we can do before we continue for example right
here we can factor out two and then we have ln2 minus ln3 right so we can use the law property
to put them together so on all this is just 2 Ln of 2 over 3. and this right here actually
can be simplified a little bit too because this is actually a difference of two squares
so we can Factor it so this becomes Ln 2 minus ln3 like so and then multiply by Ln 2 plus 3.
so in fact we are using a lot of things that we learn in algebra with this question all
right and now let's just multiply out let's just keep everything right here again so
here we have lnx square around color now an x square and then I'm just going to keep this as
how it is so we have the plus 2 L and three also 2 Ln 2 over 3 and then we have this lnx and then for
this right here we can combine them as well so I'm going to say plus and this is just Ln of 2 over 3.
and then Ln of 2 plus and three we can just multiply the inside so this is just L and six
yeah so this is the constant term and then all in all we have this being equal to zero
okay quadratic equation in terms of our next let's use the quadratic formula
all right this says L and X will be equal to negative B which is that make which is the two
Ln 2 over 3 here and then of course we have the plus well minus and let's open the square root
okay B squared we have that which is 2 Ln 2 over 3 Square minus 4 times a which is one
and then C is that so ln2 over 3 now and six okay and then everybody all over right two times one
that's the a value cool all right so let's take a look at the inside because in fact we can simplify
a little bit and let's do it right here so firstly we can square that so this is going to be 4 and
then this Square I'll just write it as Ln 2 over 3 and then Square and then minus 4 times 1 times
that so just minus four Ln 2 over 3 and then L and six and notice both of them have four and also the
Ln of two over three so that's factored out so we will have four L and two open three like so
and then here we have another right Ln two over three and a minus this right here is L and six
and this right here is what yeah a little hot minus Another Log so I can just do this divided
by that so let's see this will become our end of two over three and then divide it by six which is
the same as Ln of two over 6 times 3 right which is 18 and then that's just now end of one over
nine and that's just Ln of 9 to the negative one and put the negative to the front yeah
so this right here becomes negative ln9 and then we still have this in the front
cool all right and don't forget that we still have a square root right so it looks like hmm
okay okay and then uh is this really okay whoever negatings of the square
root and we are not looking for any complex numbers because otherwise
we we don't have that in real life unfortunately we want that to be a legitimate triangle
so don't worry don't worry in fact I'll end up two over three because two over three is less than one
this is actually negative negative times negative is positive and if you would like we can put this
negative up here and make that into exponent so we can rewrite this as square root 4 and then
Ln 3 over 2 and then Ln 9. yeah and finally yet I see an arcade to start I'm going to
write this as 2 square root of Ln is 3 over 2 l and nine all right so now what we have I'm
going to split the fractions so we have negative 2 L and 2 over 3 over 2 and then plus will minus
and then all this is just that I'm just going to write it as 2 and then square root of Ln of
3 over 2 l and nine over two aha this and that canceled this and that canceled and again let's
put a negative right here and then we can do the reciprocal very nice all right so now don't
forget that we have Ln X is equal to all this so let's see what we will actually end upwards
now here's the deal if you ignore the triangle and just look at this our next is possible to
be negative so it looks like we have two solutions however because we are talking
about a red triangle we don't want any sides to end up with a negative value therefore L next
the whole thing we don't want it to be negative so we have to get rid of that and you can check
on your own this is actually bigger than Ln of two over two so we'll have to get rid of the
negative all right and then lastly we can just do e to this power e to the power exponentiate
both sides and we will get this and not cancel so just X well let me cancel that for you guys
equals now let's do two things right here the first thing is that we are going to separate
this is just e to this power times e to the power and the second thing is that this and that will
cancel so we have three over two and then e to the power Unfortunately they cannot cancel we'll just
write it as e to the square roots of Ln of 3 over 2 times Ln 9. yep that's it cool and if you would
like you can use your calculator and let me know what the answer is I believe there's zero point
I don't remember but like you guys can go and try it so if you find that interesting then I
think you'll also like brilliant brilliant's one of the best websites for map and science end up
being a long-term user because of the variety of courses that they have currently they have
over 60 interactive courses in math science and computer science these classes are designed to
have fun and engaging I like the logic algebra and calculus classes because they often provide me a
different point of view so that I can learn something new and let me tell you they have
beginner level of math courses and also Advanced courses such as differential equations and Quantum
Computing I'm sure you will find something that you like plus if you use the link in the
description brilliant.org black and red pen then you can enjoy your 20 off of the annual premium
subscription so go ahead and check them out and thanks to brilliant for sponsoring this video