A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way. As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount, on a logarithmic scale, every unit of length corresponds to multiplying the previous value by the same amount. Hence, such a scale is nonlinear: the numbers 1, 2, 3, 4, 5, and so on, are not equally spaced. Rather, the numbers 10, 100, 1000, 10000, and 100000 would be equally spaced. Likewise, the numbers 2, 4, 8, 16, 32, and so on, would be equally spaced. Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph.
A logarithmic scale from 0.1 to 100
Semi-log plot of the Internet host count over time shown on a logarithmic scale
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
Various scales: lin–lin, lin–log, log–lin, and log–log. Plotted graphs are: y = 10x (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;
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.
A modified log transform can be defined for negative input (y<0) and to avoid the singularity for zero input (y=0) so as to produce symmetric log plots:[2][3]
for a constant C=1/ln(10).
Logarithmic units
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.
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.
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. PMC2610411. 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]. ISBN978-3-42801242-8. (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). (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.
External links
Wikimedia Commons has media related to Logarithmic scale.
logarithmic, scale, logarithmic, scale, scale, displaying, numerical, data, over, very, wide, range, values, compact, opposed, linear, number, line, which, every, unit, distance, corresponds, adding, same, amount, logarithmic, scale, every, unit, length, corre. A logarithmic scale or log scale is a way of displaying numerical data over a very wide range of values in a compact way As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount on a logarithmic scale every unit of length corresponds to multiplying the previous value by the same amount Hence such a scale is nonlinear the numbers 1 2 3 4 5 and so on are not equally spaced Rather the numbers 10 100 1000 10000 and 100000 would be equally spaced Likewise the numbers 2 4 8 16 32 and so on would be equally spaced Often exponential growth curves are displayed on a log scale otherwise they would increase too quickly to fit within a small graph A logarithmic scale from 0 1 to 100 Semi log plot of the Internet host count over time shown on a logarithmic scale Contents 1 Common uses 2 Graphic representation 2 1 Log log plots 2 2 Semi logarithmic plots 2 3 Extensions 3 Logarithmic units 3 1 Examples 3 2 Units of information 3 3 Units of level or level difference 3 4 Units of frequency interval 3 5 Table of examples 4 See also 4 1 Scale 4 2 Applications 5 References 6 Further reading 7 External linksCommon uses EditThe markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales The two logarithmic scales of a slide rule The following are examples of commonly used logarithmic scales where a larger quantity results in a higher value Richter magnitude scale and moment magnitude scale MMS for strength of earthquakes and movement in the Earth A logarithmic scale makes it easy to compare values that cover a large range such as in this map Sound level with units decibel Neper for amplitude field and power quantities Frequency level with units cent minor second major second and octave for the relative pitch of notes in music Logit for odds in statistics Palermo Technical Impact Hazard Scale Logarithmic timeline Counting f stops for ratios of photographic exposure The rule of nines used for rating low probabilities Entropy in thermodynamics Information in information theory Particle size distribution curves of soil 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 pH for acidity Stellar magnitude scale for brightness of stars Krumbein scale for particle size in geology Absorbance of light by transparent samplesSome 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 Main article Log log plot Plot on log log scale of equation of a line 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 Main article Semi log plot 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 lt 0 and to avoid the singularity for zero input y 0 so as to produce symmetric log plots 2 3 Y sgn y log 10 1 y C displaystyle Y operatorname sgn y cdot log 10 1 y C for a constant C 1 ln 10 Logarithmic units EditA 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 data storage capacity bit byte of information and information entropy nat shannon ban and of signal level decibel bel neper Logarithmic frequency quantities 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 bit byte hartley nat shannonUnits of level or level difference Edit Further information Level logarithmic quantity bel decibel neperUnits of frequency interval Edit decade decidecade savart octave tone semitone centTable of examples Edit Unit Base of logarithm Underlying quantity Interpretationbit 2 number of possible messages quantity of informationbyte 28 256 number of possible messages quantity of informationdecibel 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 intervalThe 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 Mathematics portalAlexander Graham Bell Bode plot Geometric mean arithmetic mean in logscale John Napier Level logarithmic quantity Logarithm Logarithmic mean Log semiring Preferred number Semi log plotScale Edit Order of magnitudeApplications Edit Entropy Entropy information theory pH Richter magnitude scaleReferences Edit Slide Rule Sense Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space ScienceDaily 2008 05 30 Retrieved 2008 05 31 Webber J Beau W 2012 12 21 A bi symmetric log transformation for wide range data PDF Measurement Science and Technology IOP Publishing 24 2 027001 doi 10 1088 0957 0233 24 2 027001 ISSN 0957 0233 S2CID 12007380 Symlog Demo Matplotlib 3 4 2 documentation 2021 05 08 Retrieved 2021 06 22 Further reading EditDehaene Stanislas Izard Veronique 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 Brucke 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 amp Humblot Verlag de ISBN 978 3 42801242 8 135 pages Paulin Eugen 2007 09 01 Logarithmen Normzahlen Dezibel Neper Phon naturlich 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 Wikimedia Commons has media related to Logarithmic scale GNU Emacs Calc Manual Logarithmic Units Gnu org Retrieved 2016 11 23 Non Newtonian calculus website Retrieved from https en wikipedia org w index php title Logarithmic scale amp oldid 1135043183, wikipedia, wiki, book, books, library,
