# 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 |