A refresher on logs

Only base 10 logarithms are used in this lab. Log10 x is equal to the number to which you have to raise 10 in order to get x. For example,

General rules are given at the end of the page.

What is a logarithmic scale?

Logarithmic scales are based on multiplication rather than addition. In a linear scale, the steps are evenly spaced. In a logarithmic scale, the steps increase or decrease in size.

Linear scale:

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 scale:

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.

Why do we use logarithmic scales?

Logarithmic scales allow one to examine values that span many orders of magnitude without losing information on the smaller scales. Astronomy deals with the vast and the microscopic, the bright and the dim, the strong and the weak. Logarithmic scales are often used in astronomy.

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 103
Office, classroom 50 dB 105
Normal conversation (at 1 m) 60 dB 106
City street, very busy traffic 70 dB 107
Noisest spot at Niagara Falls 85 dB 3x108
Jackhammer (at 1 m) 90 dB 109
Rock group 110 dB 1011
Threshold of pain 120 dB 1012
Jet engine (at 50 m) 130 dB 1013
Saturn rocket (at 50 m) 200 dB 1020
* The sound levels are relative to Io = 10-12 Watts/meter2.

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

b = 10 log10 (I / Io),

where I is the intensity of the sound, and Io is a reference intensity which produces a sound level of 0 dB (Io=10-12 Watts/meter2).

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

b2 - b1 = 10 log10 (I2 / I1).

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

logA N = logB N / logB A.

General rules

If AP = N, where A does not equal 0 or 1, then P = logA N.
Note that 10log10 N = N

logA (MN) = logAN + logAM

logA (M/N) = logAN - logAM

logA MP = p logA M

logA N = logB N / logB A
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