Assume that $\mathrm P(X_n=1)=x$ and $\mathrm P(X_n=0)=1-x$, hence $\mathrm E(X_n)=x$ for every $n$, and let $H_n=\sum\limits_{k=1}^n\frac1k$ denote the $n$th harmonic number, hence $H_n=\log(n)+\gamma+o(1)$.
Then $S_n-xH_n$ converges almost surely and in $L^2$ to an almost surely finite centered random variable $Y$ with variance $\mathrm E(Y^2)=x(1-x)\frac{\pi^2}6$.
In particular, $\frac1{\log n}S_n\to x$.
To see this, consider $Y_n=\sum\limits_{k=1}^n\frac1k(X_k-x)$. The random variables $X_k-x$ are centered, square integrable with variance $x(1-x)$, and independent. Hence, for every $n$, $Y_n$ is centered with variance $\sum\limits_{k=1}^n\frac1{k^2}\mathrm E((X_k-x)^2)=\sum\limits_{k=1}^n\frac1{k^2}x(1-x)$. The series $\sum\limits_k\frac1{k^2}x(1-x)$ converges hence $(Y_n)$ converges in $L^2$. Since $(Y_n)$ is the sequence of the partial sums of some independent random variables, a result due to Paul Lévy ensures that $(Y_n)$ converges almost surely as well.