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y^x = x^y


fafalone

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psi20 said in post # :

Can you give me the formula to calculate the cubic root? When I tried to make the formula, I had x = cuberoot( something with x etc etc) so it wasn't really a formula.

 

Looking in the link I posted would be a good start.

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  • 1 month later...

anyway. i did this question for integer solution of a^b=b^a given 0<a<b

 

here it is

 

take a function f where f (x)=ln(x)/x

therefore a^b=b^a if and only if f(a)=f(b). now differentiating f(x) u will find that there is a maximum at x=e. therefore the function f is increasing for x<e and decreasing for x>e. therefore its not possible for both a,b <e or a,b >e cos then f(a) will never be equal to f(b) for a<b

so the only solutions possible is a<e and b>e . now we have 0<a<e. and a is an integer. therefore a=1 or a=2. putting in a=1 we get b=1 which is not a solution as we need a<b. the only other soultion is a=2. puting in a=2 and solving f(b)=f(2) by similar method or however u want we get b =4 . therefore the only solution to a^b=b^a for 0<a<b is a=2,b=4

 

i dont think its possible to find a general solution for x^y=y^x. something do with trascendental numbers and fucntions

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It's like asking why |x| always equals a positive.

 

i dont wanna be picky, but just to correct that statement

 

|x| is non-negative. it doesnt have to be always equal to positive because it can also take the value of 0.

 

many pple confuse positive with non-negative when claiming x^2 is always positive and such.

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  • 2 weeks later...

I know this topic hasn't bee touched for a couple weeks, but there's something interesting that I found. I'm sure you guys have drawn a graph and noticed this, but just in case...

 

I drew the graph in a trail version I downloaded of a program called Autograph (we use it at my college). If you put the graph of y^x = x^y into it, you get a graph with the line y = x (not including the origin, I would assume) but it has a second curve on it. The curve crosses the points (2, 4) and (4, 2), and interestingly, it crossed the line y = x at (e, e). You probably all know this so I'm not contributing anything (I read this thread before and didn't see any mention but I may have missed it), but that suggests something about the equation of the second curve, if there is a separate equation for it. I'm only even making a bother over this because this was an equation I saw ages ago (from this thread, actually!) and I'm just curious. It's the first thread I really read on these forums. :)

 

I'd post a link to a picture of it, but I can't find any good web picture hosts. :/

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Wow, thanks. :) Hopefully this worked then:

Notice how the line on the right splits into pixels... not exactly sure what's happening here. :confused:

It almost looks like the second curve has asymptotes at about e/2

yxxyfull.JPG

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