I'm reading through Introduction to Probability by Blitzstein and on the chapter on Universality of the Uniform, there's an example where he starts with a uniform r.v. and generates a logistic r.v..
However, I'm having trouble understanding how he inverted the CDF to get $\log(\frac{u}{1-u})$:
$F^{-1}(u)=x$ iff $F(x)=u$ iff $\frac {e^{x}} {1+e^{x}}=u$. This can be written as $u(1+e^{x})=e^{x}$ or $u=e^{x} (1-u)$. This means $e^{x}=\frac u {1-u}$ or $F^{-1}(u)=x=\log (\frac u {1-u})$