Logarithmic scale


A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences between the magnitudes of the numbers involved.

Semi-log plot of the Internet host count over time shown on a logarithmic scale

Unlike a linear scale 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.

A logarithmic scale is nonlinear, 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., 10^1, 10^2, 10^3, 10^4, 10^5) and 2, 4, 8, 16, and 32 (i.e., 2^1, 2^2, 2^3, 2^4, 2^5).

Exponential growth curves are often depicted on a logarithmic scale to prevent them from expanding too rapidly and becoming too large to fit within a small graph.

A logarithmic scale from 0.1 to 100
The two logarithmic scales of a slide rule

Common uses edit

The markings on slide 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:

A logarithmic scale makes it easy to compare values that cover a large range, such as in this map.
Map of the solar system and distance to Alpha Centauri using a logarithmic scale

The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:

Some of our senses operate in a logarithmic fashion (Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of hearing 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]

Graphic representation edit

Various scales: lin–lin, lin–log, log–lin, and log–log. Plotted graphs are: y = 10 x (red), y = x (green), y = loge(x) (blue).

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 1,000.

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:

  • covers a large range of values, since the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size;
  • may contain exponential laws or power laws, since these will show up as straight lines.

A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool.

Log–log plots edit

A log-log plot condensing information that spans more than one order of magnitude along both axes

If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot.

Semi-logarithmic plots edit

If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.

Extensions edit

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 constant C=1/ln(10).

Logarithmic units edit

A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm 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 edit

Examples of logarithmic units include units of information and information entropy (nat, shannon, ban) and of signal level (decibel, bel, neper). Frequency levels or logarithmic frequency quantities have various units are used in electronics (decade, octave) and for music pitch intervals (octave, semitone, cent, etc.). Other logarithmic scale units include the Richter magnitude scale point.

In addition, several industrial measures are logarithmic, such as standard values for resistors, the American wire gauge, the Birmingham gauge used for wire and needles, and so on.

Units of information edit

Units of level or level difference edit

Units of frequency level edit

Table of examples edit

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 any power quantity (sound power, for example) sound power level (for example)
decibel 10(1/20) ≈ 1.122 any root-power quantity (sound pressure, for example) sound pressure level (for example)
semitone 2(1/12) ≈ 1.059 frequency of sound pitch interval

The two definitions of a decibel are equivalent, because a ratio of power quantities is equal to the square of the corresponding ratio of root-power quantities.[citation needed]

See also edit

Scale edit

Applications edit

References edit

  1. ^ "Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space". ScienceDaily. 2008-05-30. Retrieved 2008-05-31.
  2. ^ Webber, J Beau W (2012-12-21). "A bi-symmetric log transformation for wide-range data" (PDF). Measurement Science and Technology. 24 (2). IOP Publishing: 027001. doi:10.1088/0957-0233/24/2/027001. ISSN 0957-0233. S2CID 12007380.
  3. ^ "Symlog Demo". Matplotlib 3.4.2 documentation. 2021-05-08. Retrieved 2021-06-22.

Further reading edit

  • Dehaene, Stanislas; Izard, Véronique; Spelke, Elizabeth; Pica, Pierre (2008). "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures". Science. 320 (5880): 1217–20. Bibcode:2008Sci...320.1217D. doi:10.1126/science.1156540. PMC 2610411. PMID 18511690.
  • Tuffentsammer, Karl; Schumacher, P. (1953). "Normzahlen – die einstellige Logarithmentafel des Ingenieurs" [Preferred numbers - the engineer's single-digit logarithm table]. Werkstattechnik und Maschinenbau (in German). 43 (4): 156.
  • Tuffentsammer, Karl (1956). "Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen" [The decilog, a bridge between logarithms, decibel, neper and preferred numbers]. VDI-Zeitschrift (in German). 98: 267–274.
  • Ries, Clemens (1962). Normung nach Normzahlen [Standardization by preferred numbers] (in German) (1 ed.). Berlin, Germany: Duncker & Humblot Verlag [de]. ISBN 978-3-42801242-8. (135 pages)
  • Paulin, Eugen (2007-09-01). Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! [Logarithms, preferred numbers, decibel, neper, phon - naturally related!] (PDF) (in German). Archived (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.

External links edit

  • "GNU Emacs Calc Manual: Logarithmic Units". Gnu.org. Retrieved 2016-11-23.
  • Non-Newtonian calculus website