- Log
_{10}1 = 0, because 10^{0}= 1. - Log
_{10}10 = 1, because 10^{1}= 10. - Log
_{10}100 = 2, because 10^{2}= 100.

- Linear scales are based on addition. Consider the linear sequence
1, 3, 5, 7, 9. - To get the next number in the sequence, you
**add**2 to the previous number.

- Logarithmic scales are based on multiplication (or division).
Consider the logarithmic sequence
2, 4, 8, 16, 32. - To get the next number in the sequence, you
**multiply**the previous number by 2. We would say that this sequence represents ``base 2.'' - The sequence
1, 10, 100, 1000, 10000 - represents ``base 10,'' because you get the next term in the
sequence by multiplying the previous term by 10.

Many aspects of nature are logarithmic. The human eye responds to
changes in light intensity on a logarithmic scale. Since the
difference in light intensity between sunlit areas and shade is so
great, if your eyes did not work on a logarithmic scale, you would
never be able to discern details in the shade! The intensity of the
light from stars is often described using the **magnitude scale**,
which is a logarithmic scale relating large changes in light intensity
to the response of the human eye.

A more familiar example is the logarithmic response of the human ear to changes in sound intensity. No doubt you know that the sound level of your stereo is measured in decibels (dB). The decibel is a logarithmic unit designed to reflect the response of the human ear. Each additional decibel represents a factor of 1.26 in sound intensity. Intensity is the power transmitted per unit area; you have to increase the power output of your stereo by a factor of 1.26 for every additional decibel. What does this mean? Consider the following table:

Sound levels (b)^{*} and relative intensity (I) | ||
---|---|---|

b |
I | |

Threshold of hearing | 0 dB | 1 |

Rustle of leaves | 10 dB | 10 |

Average whisper (at 1 m) | 20 dB | 100 |

City street, no traffic | 30 dB | 10^{3} |

Office, classroom | 50 dB | 10^{5} |

Normal conversation (at 1 m) | 60 dB | 10^{6} |

City street, very busy traffic | 70 dB | 10^{7} |

Noisest spot at Niagara Falls | 85 dB | 3x10^{8} |

Jackhammer (at 1 m) | 90 dB | 10^{9} |

Rock group | 110 dB | 10^{11} |

Threshold of pain | 120 dB | 10^{12} |

Jet engine (at 50 m) | 130 dB | 10^{13} |

Saturn rocket (at 50 m) | 200 dB | 10^{20} |

Decibel levels from Halliday, Resnick and Walker (1993); and Halliday and Resnick (1986).

The decibel scale seems to describe the relative loudness of the sounds described in the table. But look at how the intensity levels change! While we may perceive a normal conversation to be about twice as loud as a city street with no traffic, the difference in the intensity of the sound is a factor of 1000. The difference in sound intensity between a jackhammer 1 meter away and the rustling of leaves is a factor of 1,000,000,000! If our ears did not work logarithmically, we would be able to distinguish differences in volume in only the very loudest sounds. We would lose many of the subtleties in music. How else might we be affected?

For those who would like to see the equations, the sound level
**b** is given by

where **I** is the intensity of the sound, and **I _{o}**
is a reference intensity which produces a sound level of 0 dB
(I

The difference in sound intensities can thus be described by the equation

The math aficionados will notice that the decibel is really defined in base 1.26. So why do the equations show logs in base 10? Normally we use base 10 logarithms or natural logarithms (base e, or 2.71) by convention. In general, the conversion for logarithms from base B to base A is given by

- If A
^{P}= N, where A does not equal 0 or 1, then P = log_{A}N. - Note that 10
^{log10 N}= N - log
_{A}(MN) = log_{A}N + log_{A}M - log
_{A}(M/N) = log_{A}N - log_{A}M - log
_{A}M^{P}= p log_{A}M - log
_{A}N = log_{B}N / log_{B}A

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