Any monotonic transformation of a utilty function will leave preferences intact. Let V = Ln(U) = (x1+ln(x2))^(1/3) now, let Z = ln(V) = (1/3(ln(x1 +ln(x2))
MU (x1) = (1/3)*(1/(1+x1 +ln(x2))
MRS = x2 = p1/p2
so p2x2= p1 M= p1x1 +p2X2 or M= p1x1+p1 or x1= (m-p1)/p1 or M/p1 -1 for M.=>p1.. Substituting x1 back into the budget constraint we see x2 =p1/p2 and is independent of income.