Discussion
A signal obeys an inverse fourth power law. By how much does it fall when the range doubles?
Answer: A
A fourth power law gives a factor of sixteen in power for a doubling of distance:
\[ 10\log_{10} 16 \approx 12 \]
so the signal falls by 12 dB, against six for free space. That much steeper roll off is what limits terrestrial coverage and is one reason cellular systems reuse frequencies at modest distances. Being able to convert quickly between a power law exponent and decibels per octave of range is the most useful arithmetic in coverage planning.
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