Logarithm formula as ratio of two sequences

Clear[n, x];
x = 10;
n = 100;
N[n*Sum[Sum[(-1)^j*(k - j)^(n - 1)*Binomial[n + 1 - 1, j], {j, 0, k}]*
     x^(k - 1)/(x - 1)^(n - 1), {k, 1, n - 1}]/
   Sum[Sum[ (-1)^j*(k - j)^n*Binomial[n + 1, j], {j, 0, k}]*
     x^(k - 1)/(x - 1)^n, {k, 1, n}], 30]
N[Log[x], 30]