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

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.


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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