In formulae for sound intensity and sound pressure, there is a logarithm part for the logarithmic increase of the quantity we're studying, and also a numerical factor 10 or 20 depending on the formula: $L=10(or\ 20)\log\frac{I}{I_0}$

If the crux/intuition of the formula is already summed in the logarithm part, what can we use the 10 or 20 for?

I thought of it as to "normalize" the obtained number (for instance 10log(2)) to human understanding (like 3dB instead of 0.3db without the 10), but it seemed too anthropic?


Actually, the reason is anthropic!

The original unit for sound intensity is the bel, named after Alexander Graham Bell. But the bel is "inconveniently large" for most purposes1, so we use the decibel (literally 1/10 of the bel or 10 dB = 1B), hence the factor 10.

Ordinary conversation is $\sim$65 dB, this would be 6.5 B


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