I am not sure how to distinguish whether a function is homothetic.
For example,
Q1. $$ U(x,y) = a\log(x) + b\log(y), $$
Q2. $$ U(x,y) = \exp[(x^a+by^a)^3 +r] $$
I can understand that these two functions are not homogenous. But i don't know why these are homothetic. Can any body explain to me??
A function is homogenous of order $k$ if $$ f(tx, ty)=t^kf(x, y). $$ A function is homothetic if it is a monotonic transformation of a homogenous function (note that this second function does not need to be homogenous itself).
Consider now the function: $$ f(x, y)=x^ay^b $$
This is homogenous, since $$ f(tx, ty)=(tx)^a(ty)^b=t^{a+b}x^ay^b=t^{a+b}f(x, y). $$
Consider now the function $$ g(z)=\log z $$ which is monotone. We have $$ g(f(x, y))=\log(f(x, y))=\log(x^ay^b)=a\log x+b\log y $$ which is your first function. This is a monotone transformation of a homogenous function, so it is homothetic.
Consider now $$ f(x, y)=x^a+by^a $$ which is homogenous since $$ f(tx, ty)=(tx)^a+b(ty)^a=t^a(x^a+by^a)=t^af(x, y). $$
Let $$ g(z)=\exp(z^3+r) $$ whose derivative is $$ g^\prime (z)=3z^2 \exp(z^3+r) $$ which is positive other than at the isolated point $z=0$, so the function $g$ is monotone. So it then follows that $$ g(f(x, y))=\exp[(f(x, y))^3+r]=\exp[(x^a+by^a)^3+r]. $$ And hence, the function you provided is a monotonic transformation of a homogenous function, meaning that it is homothetic.
