Feynman and the logarithm
You can multiply $r$ by $(1+2^{-k})$ by adding $r$ to $r$ shifted right by $k$ positions. To find the factors of a number $s$, you can start setting $r:=1$ and for $k=0,1,2,\ldots$, test if $r(1+2^{-k})\leq s$. If the test succeeds, set $r:=r(1+2^{-k})$. Along the way you can calculate $\log s$ by summing $\log 1+2^{-k}$ for every factor found.