Question about relation between powers and distance

[arthurav]

I have this formula:

(x+α)^(x+α)*(y+β)^(y+β)*(z+γ)^(z+γ)=x^x*y^y*z^z

where α+β+γ=0

I think there might be a distance relationship between x, y and z that would satisfy the equation: if x,y and z are varied with keeping a certain measure of distance constant, the equation will be verified. This is a presupposition of mine I would like to prove or disprove, but I don't know where to start.

Do you have any ideas?

Thank you.

 

[chiro]

I have this formula:

(x+α)^(x+α)*(y+β)^(y+β)*(z+γ)^(z+γ)=x^x*y^y*z^z

where α+β+γ=0

I think there might be a distance relationship between x, y and z that would satisfy the equation: if x,y and z are varied with keeping a certain measure of distance constant, the equation will be verified. This is a presupposition of mine I would like to prove or disprove, but I don't know where to start.

Do you have any ideas?

Thank you.

Hey arthurav and welcome to the forums.

One suggestion I have is to take logs of both sides and then take it from there. You should get things in terms of xlog(x) + ylog(y) + zlog(z) = (x+a)log(x+a) + (y+b)log(y+b) + (z+c)log(z+c).

Also, in addition this might help: http://www.arts.cornell.edu/econ/arazin/loglinearization.pdf