The problem is to show (a) that ln(x) is not a ratio of two polynomials, i.e. that q(x)ln(x) + p(x) = 0 implies that q(x) and p(x) are both identically zero, and (b) that a[m](x) ln(x)^m + ... + a(x) = 0 likewise implies that the a[k]'s are all zero.
The key idea is to differentiate q(x)ln(x) + p(x) enough times so that the coefficient of ln(x) is zero. Then you can work your way back using polynomial solutions of linear differential equations.
This idea works for part (b) of Question (1) as well. Note also that part (a) is contained in part (b); if you can prove part (b), you've proved part (a).