As Part 1 introduces, we get decibel using the formula

$$ l = 10 \times log \frac{p_1}{p_2} (dB) $$

I'd blame my poor English ability that I didn't realise what the name *decibel* really meant for quite a few years. It's called deci-bel so apparently there's a *Bel* first.

The bel (B) and the smaller decibel (dB) are units of measurement of sound pressure level (SPL) invented by Bell Labs and named after him

from: Wikipedia

That's him! As stated, using *deci-*, people can make *bel* smaller, and meanwhile we have to add the \(10 \times \) in the formula, to make *bel* become *decibel*.

You might notice that when talking about voltage and current, the formula becomes

$$ 20 \times log \frac{p_1}{p_2} (dB) $$

So why 20 here? I was very confused. I asked around and got this answer: For measuring something like a **magnitude**, we use \(20 \times \), otherwise for the **power** stuffs, use \(10 \times \)... Okay then it's time to test my English again: what kind of thing is a **magnitude**? And what is not? I reckon it's a correct but not so good answer.

Finally I met a guy, he took out a piece of paper and wrote some high school maths on it:

$$ P=U \times I=U \times (U/R) = U^2 / R $$

Yeah that's Ohm's Law, I get it. Then if \(P_r\) and \(U_r\) are the references and \(U\) is the voltage we are measuring, from the original \(10 \times \) formula we have:

\[ 10 \times log (P / P_r) \]

\[ = 10 \times log ((U^2/R)/(U_r^2/R)) \]

\[ = 10 \times log (U/U_r)^2 \]

\[ = 2 \times 10 \times log (U/U_r) = 20 \times log (U/U_r) \]

That's how \(20 \times \) comes up. Same process for the current (Use \(P=I^2 \times R\)). Maths can be scary, but useful.