Cobb-Douglas and Logarithm Utility Functions
Utility functions are invariant with respect to positive monotonic transformations (PMT).
Take $U(x,y)=x^\alpha y^{1-\alpha}$, and let $V(x,y)=\log(U(x,y))$ be a PMT of $U$. 
Thus $V$ and $U$ both represent the same preference, and thus demand functions for $x$ and $y$ are the same.