# Online Tone Generator

What is a decibel?

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 $10\log_{10} \left[\frac{2I_0}{I_0}\right] = 10\log_{10}2 = 3 dB$

Scenario Number of times more intense than I0 Intensity in dB
Limit of human hearing 1 0
Sound of Breathing 10 10
Whispering 100 20
Loud Conversation 100,000 50
Busy Street 10,000,000 70
Thunder 10,000,000,000 100
Jet Plane on Take-off 100,000,000,000,000 140
Space Rocket Launch 10,000,000,000,000,000,000 190