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The internet is full of overly complicated explanations of this widely used but rarely understood unit of measurement.

Firstly, it is important to understand exponents and logarithms. Here's a simple example: \[ 100 = 10^2 = 10\times10 \] Here, the little 2 is called the exponent. It means multiply 10 by itself twice. Similarly, for: \[ 1000 = 10^3 = 10\times10\times10 \] 3 is the exponent, and means multiply 10 by itself 3 times.

A logarithm is the exact opposite of an exponent. Take another simple example. This is pronounced "the logarithm to base ten of one thousand equals three": \[ \log_{10} 1000 = 3 \] 10 is called the base. The logarithm of 1000 is simply the number of times the base (in this case 10) must be multiplied by itself to get to 1000, i.e. 3 times. And you should notice that this is just the same as the exponent from the equation above. Each equation uses the same three numbers, just arranged in a slightly different way.

Now, the strange thing is that exponents and bases don't have to be whole numbers. In fact, they can be any numbers at all. Here is a completely random example: \[ 6^{4.7} = 4542.6674 \] \[ \log_{6} 4542.6674 = 4.7 \] One important rule to learn is that the logarithm of 1 is always zero, regardless of its base.

The decibel

The decibel (dB) is one tenth of a bel and is named in honour of Alexander Graham Bell. The intensity in dB of a sound of intensity I is given by \[ I(dB) = 10\log_{10}\left[\frac{I}{I_0}\right] \] It is a relative unit which means it only makes sense when you have a reference level to compare to. For our reference we use the threshold of human hearing. The quietest thing any human can hear, I0, is one trillionth of a watt per square meter. So, if a sound is twice as intense as I0, its intensity will be \[ 10\log_{10} \left[\frac{2I_0}{I_0}\right] = 10\log_{10}2 = 3 dB \]