Let $X=(X_1,X_2,...,X_n)$ be a random sample from the logistic distribution with cdf $F(x)=1/(1+e^{-x})$, $x \in \Bbb R.$ Let $Y_n$ be the $n$-th order statistic. Find the limiting distribution for $Y_n-\ln(n)$ as $n \to \infty $.
Here's what I've done so far:
$$F_{Z_n}=P(Z_n\le z) \\=P(Y_n\le z+\ln(n)) \\=F_{Y_n}(z+\ln(n)) =F^n(z+\ln(n));\\ F(z+\ln(n))=1/({1+e^{-(z+\ln(n))}})$$
Now how do I find $F^n(z+\ln(n))$?