I was using decibels for a few years before I understood them properly. When I was in college, we used decibels a lot to describe the gain of an amplifier, e.g. "The gain of this amplifier is 45 dB". I had the written formulae for converting to and from decibels, but I never really understood them. It's a shame that so many institutions only give a very limited understanding of decibels, when in actual fact you can get a complete understanding of it if you sit down for a few minutes.
And when you understand something fully, you actually don't need the mathematical formulae because you can figure it out all by yourself. For instance if I tell you that an amplifier has a gain of 45 dB, then can you tell me what the actual gain is (i.e. give me an actual decimal number instead of decibels)? Well first of all, we know that a decibel is a tenth of a bell, so:
45 decibel = 4.5 bel
Next, we know that a bel is simply 10 to the power of X:
4.5 bel = 10 to the power of 4.5
So the gain of the amplifier is:
10 to the power of 4.5
which is going to be somewhere between 10000 (four zeroes) and 100000 (five zeroes). Using a calculator, I get 31622.776601684. So if the input to the amplifier is 1 volt, the output is gonna be 31.6 kilovolts. So there you go, no need for fancy formulae.
And as I showed with my "I have 10 decibel dogs" example, you can use decibels for counting anything. Instead of sending your nephew a cheque for 1 million dollars, send him a cheque for 60 decibel dollars. The maths is easy:
We start of with 1 million dollars: 1000000
One million has six zeroes, so that's 10 to the power of 6. Instead of saying 10 to the power of 6, we can simply say 6 bels. Next we want it in decibels, which are "tenths of a bell", so we simply multiply it by 10, giving 60 decibels. So 1 million dollars is 60 decibel dollars.
When your nephew receives the cheque, the maths he does is:
1) Well 60 decibels = 6 bels, that's easy
2) Next, instead of saying 6 bels, he says 10 to the power of 6
3) Then he just writes 1 with 6 zeroes after it, giving 1 million
Things are a tiny bit more complicated if you start off with a number such as 2 million dollars. You need to find out what X is in the following equation:
2 million = 10 to the power of X
When we dealt with numbers such as ten, or one hundred, or one million, it was easy because all we had to do was count the zeroes. However if the number you start off with isn't a 1 followed by zeroes, then you need to do some calculation. In order to solve this equation:
2000000 = 10 to the power of X
You re-write it as:
X = log10 (2000000)
That's what logarithmic functions are for, they're for finding out what "to the power of X" is.
So in this case, for 2 million dollars, using a calculator to find log10 of 2 million, I get X = 6.301029996. So that's 6.3 bels. To get decibels, just multiply by 10, giving 63 decibels. So there you have it, 2 million dollars is 63 decibel dollars.