`f(x,y) = 2ln(x/y)` Determine whether the function is homogenous and if it is, determine its degree

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A function `f(x, y)` is called homogenous (homogeneous) of degree `n,` if for any `x,` `y` we have `f(tx, ty) = t^n f(x, y).`

The given function is homogenous of degree 0, because

`f(tx, ty) = 2ln((tx)/(ty)) = 2ln(x/y) = f(x, y) = t^0 f(x,y).`

The difficulty is that this function is not defined for all `x` and `y.` The above equality is true for all `x` and `y` for which it has sense.

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Given

`f(x,y)=2ln(x/y)`

if the function has to be homogenous then it has to be of the form

`f(tx,ty)=t^n f(x,y)`

so,

`f(tx,ty)=2ln((tx)/(ty))= 2ln(x/y)` as , on cancelling t .

so the function is of the form `f(tx,ty)=t^n f(x,y)` and the degree is n=0

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