How is a decibel a tenth of a Bel? By definition, one Bel is log10(P2/P1) (in the power example) and one dB is 10log10(P2/P1), so it appears to me that one dB is ten Bels, not one-tenth of a Bel.  Yet, many online publications, university publications, and others all seem to agree that one dB is one-tenth of a Bel.  How do they figure?  (And please don't say anything about the definition of the deci- prefix, which only clouds the explanation.)

Thanks!
Noji Ratzlaff
Orem, Utah
One Bel is not $\log_{10}(P_2/P_2)$ (especially as this depends on $P_1$ and $P_2$). Rather, the number of Bels by which $P_2$ differes from $P_1$ is obtained by computing $\log_{10}(P_2/P_2)$. And because one dB is one tenth of one B, the number of decibels is ten times as large.

This is the same as that the number of decimeters is obtained by multiplying the number of meters that make a distance by $10$: I am $1.85$ meters tall, so I am $18.5$ decimeters tall (or $185$ centimeters, or $0.00185$ kilometers)
A bel is defined by using the formula: $$\log (x÷y)$$
So, if we compare 2 power ratios... Say 10 and 1; then our Bel = log (10÷1) = log (10) = 1
Similarly, if we compare 100 and 1; then our Bel = log (100÷1) = log (100) = 2
A Decibel is 1/10th of a Bel. Which means, there are 10 Decibels in 1 Bel.
So for the same ratios...
10:1 = 1 Bel OR 10 decibels;
100:1 = 2 Bels OR 20 decibels; and
1000:1 = 3 Bels OR 30 decibels.
As you see, for every Bel, we get an equivalent value of 10 Decibels.
Meaning…
If our Bel value = 1, our dB value = 10.
If our Bel value = 2, our dB value = 10 x 2 = 20.
If our Bel value = 3, our dB value = 10 x 3 = 30.
Which means, in other for us to get our Decibel value, we need to multiply our Bel value by 10.
So effectively, if we were to compute a formula for decibels directly, we can say that...
dB = 10 x log (x÷y)