## What is a Decibel? — Logarithms & Math

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In the previous post, we learned that the decibels belongs to a group of units called a relative units. This means that decibels are always expressed as a ratio of a measured value to a known reference value. There is, of course, a bit more to decibels than dividing one number by another, but don’t worry—the math isn’t too bad. We’ll walk you through it nice and slow.

So, why bother with more math in the first place? Shouldn’t it be enough to measure the ratio with respect to the reference?

That would be a good idea, except for a couple of issues. The first is that quantities like sound pressure levels can span a large range—pressure levels might be anywhere from 0.00002 to 200 pascals. That range of numbers is simply too large to be useful. (People like to read numbers in the 1 to 100 range because they are most commonly used.) More importantly, how do you know what a given sound pressure level sounds like to the human ear? Is it quiet or loud? And how can you represent such a large range of values on a graph? It would be quite difficult.

To remedy this problem, the people at the Bell System (the ones who invented the decibel) back in 1923 used a mathematical operation called a logarithm to compress the ratio into more reasonable numbers.

A logarithm gives you the exponent you need to reach a certain number by repeatedly multiplying a base number by itself. For example, if you want to represent 100 with a logarithm and your base is 10, you need to raise 10 to a power of 2 (or multiply 10 two times: 10 x 10) to reach 100. Thus, the logarithm base 10 of 100 is 2. In mathematical notation, this process looks like this: The scientists at Bell Laboratories took the logarithm of the ratio of the measured value to the reference value inside the parentheses to make the relative values more manageable. They then added a multiplicative scaling factor to the front of the logarithm to fine tune the range of the decibel, and the full mathematical expression for the decibel was born: Note that this expression is for root-power quantities. If you are using power quantities, the expression is slightly different: This fully developed decibel scale offers several distinct benefits over absolute measurements. From a practical standpoint, the full range of pressures from just perceptible to the threshold of pain for human hearing can be represented with values from 0 to 140, which is much better than trying to compare 0.00002 to 200 Pa.  Also, it turns out that human hearing is generally logarithmic in nature, so decibels match well with human perception and experience. The decibel also allows quantities—like sound pressure levels—to be graphed and analyzed more easily.

There is, however, one major pitfall to watch out for when working in decibels. Decibels cannot be directly added or subtracted. All relative units, including the decibel, must be taken back to absolute units to be added or subtracted. This is true even for decibel values with identical references. You have to return to absolute units to do math on a decibel quantity. If it helps, think of decibels like a presentation tool, not as measured values.

There you have it. Decibels are not as confusing as they might appear at first glance. They are a clever and useful tool for making sense of the world of acoustics. 