= Cyclic logarithmic coloring
{title2=$\lfloor\log_b x\rfloor\bmod q$}
For $b>1$ and an <integer> $q\geq2$, assign $x\geq1$ the color $\lfloor\log_b x\rfloor\bmod q$. 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 $\{1,\ldots,q-1\}$, 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.
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