How is a decibel a tenth of a Bel?

Question

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

So, your example in parentheses shows that 10 Bels is one decibels, contradicting the statement "You need 10 dB to make one Bel" Is that not correct? — Noji, May 10 '15 at 13:34
When we express your height (say 8 feet) in feet vs inches. your height is 8 feet = 96 inches. So the number to represent your height in inches is $12$ times of that in feet. However, the unit itself "inch" is only 1/12 of "foot. The same thing happens to "bel" vs "dbel", if your express some intensity in "bel" vs "dbel", the number to present the intensity in "dbel" is $10$ times of that in "bel" but the unit itself "dbel" is only 1/10 of "bel". — achille hui, May 10 '15 at 13:40
I see...it's a conversion factor, not really an equation. Thx! — Noji, May 10 '15 at 20:03

score 2 · Answer 1 · answered May 10 '15 at 13:37

2

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)

answered May 10 '15 at 13:37

Hagen von Eitzen

374,180

1

Ahh...the centimeters example really helped! It's the number of that makes the difference. In other words, even though 1 centimeter is 1/100 of a meter, it takes 100 times the number of meters to get number of centimeters. Any by the same token, even though the dB is 1/10 of a Bel, it takes 10 times the number of Bels to get the number of decibels. Thx! – Noji May 10 '15 at 20:00
The final conclusion is (by definition) 1 Bel = log10(P2/P1) and (by definition) 1dB = 10log10(P2/P1), so therefore 1 dB = 10 Bels. So those who assert that 1 dB is a tenth of a Bel are apparently in error. – Noji May 09 '17 at 22:19

score 0 · Answer 2 · answered Oct 08 '17 at 16:19

0

because A deciBel is equal to deci=1/10 of a Bel => B=1/10dB Now, if this equation multiply x10 from both side => 10B=dB Therefore when we have 5B it must be 50dB because dB is 10 times smaller than B.

Mehrdad

answered Oct 08 '17 at 16:19

user489084

1

score -1 · Answer 3 · edited Sep 17 '20 at 10:27

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)

How is a decibel a tenth of a Bel?

3 Answers3