Cyclic logarithmic coloring ($\lfloor {\log }_{b}x\rfloor \mathrm{mod}q$)

For $b>1$ and an integer $q\ge 2$, assign $x\ge 1$ the color $\lfloor {\log }_{b}x\rfloor \mathrm{mod}q$. The floor function partitions the domain into half-open geometric bins. If $1<{\log }_b(y/x)<q-1$, their bin indices differ by an integer in $\{1,\dots ,q-1\}$, so their colors differ. For example, $b=3/2$ and $q=3$ separates every ratio in $[1.9,2]$. Composing this finite coloring with another logarithm can obstruct monochromatic multiplicative configurations whose logarithms have such ratios.