Alogarithmic scale (orlog scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved.
Unlike a linearscale where each unit of distance corresponds to the same increment, on a logarithmic scale each unit of length is a multiple of some base value raised to a power, and corresponds to the multiplication of the previous value in the scale by the base value. In common use, logarithmic scales are in base 10 (unless otherwise specified).
A logarithmic scale isnonlinear, and as such numbers with equal distance between them such as 1, 2, 3, 4, 5 are not equally spaced. Equally spaced values on a logarithmic scale have exponents that increment uniformly. Examples of equally spaced values are 10, 100, 1000, 10000, and 100000 (i.e., 101, 102, 103, 104, 105) and 2, 4, 8, 16, and 32 (i.e., 21, 22, 23, 24, 25).
Exponential growth curves are often depicted on a logarithmic scalegraph.
The markings onslide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.
The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
Some of oursenses operate in a logarithmic fashion (Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense ofhearing perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers in some cultures.[1]
The top left graph is linear in the X- and Y-axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y-axis of the bottom left graph, and the Y-axis ranges from 0.1 to 1000.
The top right graph uses a log-10 scale for just the X-axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y-axis.
Presentation of data on a logarithmic scale can be helpful when the data:
Aslide rule has logarithmic scales, andnomograms often employ logarithmic scales. Thegeometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmicgraph paper was a commonly used scientific tool.
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as alog–log plot.
If only theordinate orabscissa is scaled logarithmically, the plot is referred to as asemi-logarithmic plot.
A modified log transform can be defined for negative input (y < 0) to avoid the singularity for zero input (y = 0), and so produce symmetric log plots:[2][3]
for a constantC=1/ln(10).
Alogarithmic unit is aunit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of alogarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.
Examples of logarithmic units includeunits of information andinformation entropy (nat,shannon,ban) and ofsignal level (decibel, bel,neper).Frequency levels or logarithmic frequency quantities have various units are used in electronics (decade,octave) and for music pitchintervals (octave,semitone,cent, etc.). Other logarithmic scale units include theRichter magnitude scale for earthquakes and thepH value foracidity orbasicity.
In addition, some industrial measures are logarithmic, such as mostwire gauges used for wires and needles.
| Unit | Base of logarithm | Underlying quantity | Interpretation |
|---|---|---|---|
| bit | 2 | number of possible messages | quantity of information |
| byte | 28 = 256 | number of possible messages | quantity of information |
| decibel | 10(1/10) ≈ 1.259 | anypower quantity (sound power, for example) | sound power level (for example) |
| decibel | 10(1/20) ≈ 1.122 | anyroot-power quantity (sound pressure, for example) | sound pressure level (for example) |
| semitone | 2(1/12) ≈ 1.059 | frequency ofsound | pitch interval |
The two definitions of a decibel are equivalent, because a ratio ofpower quantities is equal to the square of the corresponding ratio ofroot-power quantities.[citation needed][4]
{{cite book}}:ISBN / Date incompatibility (help) (135 pages)