Integral of 1/(x*ln^p(x)) from e^2 to infinity for p>0

[bransonsb]

So I can get the indefinite integral to [Ln^1-p(x)]/(1-p) which is correct according to wolfram alpha and symbolab. I understand the limit of integration substitution stuff so you'd end up with lim_b->infinity [Ln^1-p(x)]/(1-p)|^b_e^2 but both my book and wolfram alpha say that for the definite integral, the answer is 1/[(p-1)(2^p-1)]. Given that Ln(infinity)=infinity / is undefined, why does the integral not diverge?