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Logarithmic scale
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<h2 id="common-uses">Common uses</h2> <p>The markings on <a href="/facts/Slide_rule/KzIEdWad">slide rules</a> are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales. </p><p>The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value: </p> <ul><li><a href="/facts/Richter_magnitude_scale/UDhqkAX1">Richter magnitude scale</a> and <a href="/facts/Moment_magnitude_scale/jLCldUuF">moment magnitude scale</a> (MMS) for strength of <a href="/facts/Earthquakes/NGjIzagw">earthquakes</a> and <a href="/facts/Motion_(physics)/dCnP7MlB">movement</a> in the <a href="/facts/Earth/HLLmDfj0">Earth</a></li></ul> <ul><li><a href="/facts/Sound_level_(disambiguation)/Pyof5AyU">Sound level</a>, with the unit <a href="/facts/Decibel/0LyxAPXh">decibel</a></li> <li><a href="/facts/Neper/nzHnA6ev">Neper</a> for amplitude, field and power quantities</li> <li><a href="/facts/Frequency_level/0Yt12Pfr">Frequency level</a>, with units <a href="/facts/Cent_(music)/lsm8TJgX">cent</a>, <a href="/facts/Minor_second/lVBsLYSA">minor second</a>, <a href="/facts/Major_second/x94d53Fo">major second</a>, and <a href="/facts/Octave/RA4gWvkV">octave</a> for the relative pitch of notes in <a href="/facts/Music/hDVP0xsQ">music</a></li> <li><a href="/facts/Logit/5674C2DL">Logit</a> for <a href="/facts/Odds/VtM45sI6">odds</a> in <a href="/facts/Statistics/DNPsKGYU">statistics</a></li> <li><a href="/facts/Palermo_Technical_Impact_Hazard_Scale/GXdgcHQP">Palermo Technical Impact Hazard Scale</a></li> <li>Logarithmic timeline</li> <li>Counting <a href="/facts/F-stop/QcdW0077">f-stops</a> for ratios of <a href="/facts/Photographic_exposure/Rzr0aGMd">photographic exposure</a></li> <li>The rule of nines used for rating low <a href="/facts/Probabilities/fgYvMhwb">probabilities</a></li> <li><a href="/facts/Entropy/s52IXeFz">Entropy</a> in <a href="/facts/Thermodynamics/q4hIx3yP">thermodynamics</a></li> <li><a href="/facts/Information/3LdrzXtJ">Information</a> in <a href="/facts/Information_theory/qNLH3U5P">information theory</a></li> <li>Particle size distribution curves of soil</li></ul> <p>The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value: </p> <ul><li><a href="/facts/PH/RnSv8D7y">pH</a> for acidity</li> <li><a href="/facts/Apparent_magnitude/7pCTF1Me">Stellar magnitude scale</a> for brightness of <a href="/facts/Star/PFbd6ICx">stars</a></li> <li><a href="/facts/Krumbein_scale/0jzacuSU">Krumbein scale</a> for <a href="/facts/Particle_size_(grain_size)/0jzacuSU">particle size</a> in <a href="/facts/Geology/WGvVRIYF">geology</a></li> <li><a href="/facts/Absorbance/Rnv6VMX1">Absorbance</a> of light by transparent samples</li></ul> <p>Some of our <a href="/facts/Sense/n9kYQ3rf">senses</a> operate in a logarithmic fashion (<a href="/facts/Weber%E2%80%93Fechner_law/K2dNPlbW">Weber–Fechner law</a>), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of <a href="/facts/Hearing_(sense)/EPYrVkSx">hearing</a> 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.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="graphic-representation">Graphic representation</h2> <p>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. </p><p>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. </p><p>Presentation of data on a logarithmic scale can be helpful when the data: </p> <ul><li>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;</li> <li>may contain <a href="/facts/Exponential_law/m4yEqKJ7">exponential laws</a> or <a href="/facts/Power_law/DhAjYwwz">power laws</a>, since these will show up as straight lines.</li></ul> <p>A <a href="/facts/Slide_rule/KzIEdWad">slide rule</a> has logarithmic scales, and <a href="/facts/Nomogram/ysXLNuVY">nomograms</a> often employ logarithmic scales. The <a href="/facts/Geometric_mean/6Nf6uk1A">geometric mean</a> of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic <a href="/facts/Graph_paper/NKJBaxCS">graph paper</a> was a commonly used scientific tool. </p> <h3>Log–log plots</h3> <p class="note">Main article: <a href="/facts/Log%E2%80%93log_plot/VES46X8k">Log–log plot</a></p> <p>If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a <a href="/facts/Log%E2%80%93log_plot/VES46X8k">log–log plot</a>. </p> <h3>Semi-logarithmic plots</h3> <p class="note">Main article: <a href="/facts/Semi-log_plot/H2a0ANuF">Semi-log plot</a></p> <p>If only the <a href="/facts/Ordinate/TGAvIATE">ordinate</a> or <a href="/facts/Abscissa/TGAvIATE">abscissa</a> is scaled logarithmically, the plot is referred to as a <a href="/facts/Semi-logarithmic/H2a0ANuF">semi-logarithmic</a> plot. </p> <h3>Extensions</h3> <p>A modified log transform can be defined for negative input (<i>y</i> < 0) to avoid the singularity for zero input (<i>y</i> = 0), and so produce symmetric log plots:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> Y = sgn ( y ) ⋅ log 10 ( 1 + | y / C | ) {\displaystyle Y=\operatorname {sgn}(y)\cdot \log _{10}(1+|y/C|)} <p>for a constant <i>C</i>=1/ln(10). </p> <h2 id="logarithmic-units">Logarithmic units</h2> <p>A logarithmic unit is a <a href="/facts/Units_of_measurement/pvTpWbmC">unit</a> that can be used to express a quantity (<a href="/facts/Physical_quantity/S8pWBEpK">physical</a> or mathematical) on a logarithmic scale, that is, as being proportional to the value of a <a href="/facts/Logarithm/QeXaV97q">logarithm</a> 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. </p> <h3>Examples</h3> <p>Examples of logarithmic units include <a href="/facts/Units_of_information/lxjSrVJM">units of information</a> and <a href="/facts/Information_entropy/NLg4NLvt">information entropy</a> (<a href="/facts/Nat_(unit)/UgNgfVgI">nat</a>, <a href="/facts/Shannon_(unit)/VEevYzKM">shannon</a>, <a href="/facts/Ban_(information)/HHnoxYMR">ban</a>) and of <a href="/facts/Signal_level/qohClhyG">signal level</a> (<a href="/facts/Decibel/0LyxAPXh">decibel</a>, bel, <a href="/facts/Neper/nzHnA6ev">neper</a>). <a href="/facts/Frequency_level/0Yt12Pfr">Frequency levels</a> or logarithmic frequency quantities have various units are used in electronics (<a href="/facts/Decade_(log_scale)/qEYTPWw8">decade</a>, <a href="/facts/Octave_(electronics)/uGdI87T5">octave</a>) and for music pitch <a href="/facts/Interval_(music)/aFwkkgpY">intervals</a> (<a href="/facts/Octave/RA4gWvkV">octave</a>, <a href="/facts/Semitone/lVBsLYSA">semitone</a>, <a href="/facts/Cent_(music)/lsm8TJgX">cent</a>, etc.). Other logarithmic scale units include the <a href="/facts/Richter_magnitude_scale/UDhqkAX1">Richter magnitude scale</a> point. </p><p>In addition, several industrial measures are logarithmic, such as standard values for <a href="/facts/Logarithmic_resistor_ladder/mhz9LNsX">resistors</a>, the <a href="/facts/American_wire_gauge/z4yWQfgt">American wire gauge</a>, the <a href="/facts/Birmingham_gauge/xla9Qf0U">Birmingham gauge</a> used for wire and needles, and so on. </p> <h3>Units of information</h3> <ul><li><a href="/facts/Bit/S972NSaD">bit</a>, <a href="/facts/Byte/BNHUp9Qq">byte</a></li> <li><a href="/facts/Hartley_(unit)/HHnoxYMR">hartley</a></li> <li><a href="/facts/Nat_(unit)/UgNgfVgI">nat</a></li> <li><a href="/facts/Shannon_(unit)/VEevYzKM">shannon</a></li></ul> <h3>Units of level or level difference</h3> <p class="note">Further information: <a href="/facts/Level_(logarithmic_quantity)/0Yt12Pfr">Level (logarithmic quantity)</a></p> <ul><li><a href="/facts/Bel_(unit)/0LyxAPXh">bel</a>, <a href="/facts/Decibel/0LyxAPXh">decibel</a></li> <li><a href="/facts/Neper/nzHnA6ev">neper</a></li></ul> <h4>Units of frequency level</h4> <ul><li><a href="/facts/Decade_(log_scale)/qEYTPWw8">decade</a>, <a href="/facts/One-third_octave/VExWkzkl">decidecade</a>, <a href="/facts/Savart/8oJwc8ry">savart</a></li> <li><a href="/facts/Octave/RA4gWvkV">octave</a>, <a href="/facts/Whole_tone/x94d53Fo">tone</a>, <a href="/facts/Semitone/lVBsLYSA">semitone</a>, <a href="/facts/Cent_(music)/lsm8TJgX">cent</a></li></ul> <h3>Table of examples</h3> <table><tbody><tr><th>Unit</th><th>Base of logarithm</th><th>Underlying quantity</th><th>Interpretation</th></tr><tr><td><a href="/facts/Bit/S972NSaD">bit</a></td><td>2</td><td>number of possible messages</td><td><a href="/facts/Quantities_of_information/hI8RLwh9">quantity of information</a></td></tr><tr><td><a href="/facts/Byte/BNHUp9Qq">byte</a></td><td>28 = 256</td><td>number of possible messages</td><td><a href="/facts/Quantities_of_information/hI8RLwh9">quantity of information</a></td></tr><tr><td><a href="/facts/Decibel/0LyxAPXh">decibel</a></td><td>10(1/10) ≈ 1.259</td><td>any <a href="/facts/Power,_root-power,_and_field_quantities/PEHfJtaG">power quantity</a> (<a href="/facts/Sound_power/CDs5e3ta">sound power</a>, for example)</td><td><a href="/facts/Sound_power_level/CDs5e3ta">sound power level</a> (for example)</td></tr><tr><td><a href="/facts/Decibel/0LyxAPXh">decibel</a></td><td>10(1/20) ≈ 1.122</td><td>any <a href="/facts/Power,_root-power,_and_field_quantities/PEHfJtaG">root-power quantity</a> (<a href="/facts/Sound_pressure/MgJAcp2I">sound pressure</a>, for example)</td><td><a href="/facts/Sound_pressure_level/MgJAcp2I">sound pressure level</a> (for example)</td></tr><tr><td><a href="/facts/Semitone/lVBsLYSA">semitone</a></td><td>2(1/12) ≈ 1.059</td><td><a href="/facts/Frequency/QUSbPaW8">frequency</a> of <a href="/facts/Sound/DlKXlPuF">sound</a></td><td><a href="/facts/Pitch_interval/ANuY50Nf">pitch interval</a></td></tr></tbody></table> <p>The two definitions of a decibel are equivalent, because a ratio of <a href="/facts/Power,_root-power,_and_field_quantities/PEHfJtaG">power quantities</a> is equal to the square of the corresponding ratio of <a href="/facts/Power,_root-power,_and_field_quantities/PEHfJtaG">root-power quantities</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Alexander_Graham_Bell/oQer2abq">Alexander Graham Bell</a></li> <li><a href="/facts/Bode_plot/xcF1ej98">Bode plot</a></li> <li><a href="/facts/Geometric_mean/6Nf6uk1A">Geometric mean</a> (arithmetic mean in logscale)</li> <li><a href="/facts/John_Napier/Wg0HAmq5">John Napier</a></li> <li><a href="/facts/Level_(logarithmic_quantity)/0Yt12Pfr">Level (logarithmic quantity)</a></li> <li><a href="/facts/Log%E2%80%93log_plot/VES46X8k">Log–log plot</a></li> <li><a href="/facts/Logarithm/QeXaV97q">Logarithm</a></li> <li><a href="/facts/Logarithmic_mean/j386VtQb">Logarithmic mean</a></li> <li><a href="/facts/Log_semiring/cSDL8zg6">Log semiring</a></li> <li><a href="/facts/Preferred_number/FdAmtumP">Preferred number</a></li> <li><a href="/facts/Semi-log_plot/H2a0ANuF">Semi-log plot</a></li></ul> <h3>Scale</h3> <ul><li><a href="/facts/Order_of_magnitude/cWzRP2dM">Order of magnitude</a></li></ul> <h3>Applications</h3> <ul><li><a href="/facts/Entropy/s52IXeFz">Entropy</a></li> <li><a href="/facts/Entropy_(information_theory)/NLg4NLvt">Entropy (information theory)</a></li> <li><a href="/facts/PH/RnSv8D7y">pH</a></li> <li><a href="/facts/Richter_magnitude_scale/UDhqkAX1">Richter magnitude scale</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Dehaene, Stanislas; Izard, Véronique; <a href="/facts/Elizabeth_Spelke/0wPYXHR9">Spelke, Elizabeth</a>; <a href="/facts/Pierre_Pica/A4DrnCUN">Pica, Pierre</a> (2008). <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2610411">"Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures"</a>. <i>Science</i>. 320 (5880): 1217–20. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008Sci...320.1217D">2008Sci...320.1217D</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1126%2Fscience.1156540">10.1126/science.1156540</a>. <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2610411">2610411</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/18511690">18511690</a>.</li> <li>Tuffentsammer, Karl; Schumacher, P. (1953). "Normzahlen – die einstellige Logarithmentafel des Ingenieurs" [Preferred numbers - the engineer's single-digit logarithm table]. <i>Werkstattechnik und Maschinenbau</i> (in German). 43 (4): 156.</li> <li>Tuffentsammer, Karl (1956). "Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen" [The decilog, a bridge between logarithms, decibel, neper and preferred numbers]. <i>VDI-Zeitschrift</i> (in German). 98: 267–274.</li> <li>Ries, Clemens (1962). <i>Normung nach Normzahlen</i> [<i>Standardization by preferred numbers</i>] (in German) (1 ed.). Berlin, Germany: Duncker & Humblot Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-42801242-8. (135 pages)</li> <li>Paulin, Eugen (2007-09-01). <a href="http://www.rechenschieber.org/Normzahlen.pdf"><i>Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt!</i></a> [<i>Logarithms, preferred numbers, decibel, neper, phon - naturally related!</i>] (PDF) (in German). <a href="https://web.archive.org/web/20161218223050/http://www.rechenschieber.org/Normzahlen.pdf">Archived</a> (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.</li></ul> <h2 id="external-links">External links</h2> Wikimedia Commons has media related to Logarithmic scale. <ul><li><a href="https://www.gnu.org/software/emacs/manual/html_node/calc/Logarithmic-Units.html">"GNU Emacs Calc Manual: Logarithmic Units"</a>. <i>Gnu.org</i>. Retrieved 2016-11-23.</li> <li><a href="https://sites.google.com/site/nonnewtoniancalculus/">Non-Newtonian calculus website</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space". ScienceDaily. 2008-05-30. Retrieved 2008-05-31. <a href="https://www.sciencedaily.com/releases/2008/05/080529141344.htm" target="_blank">https://www.sciencedaily.com/releases/2008/05/080529141344.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="https://kar.kent.ac.uk/32810/2/2012_Bi-symmetric-log-transformation_v5.pdf" target="_blank">https://kar.kent.ac.uk/32810/2/2012_Bi-symmetric-log-transformation_v5.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Symlog Demo". Matplotlib 3.4.2 documentation. 2021-05-08. Retrieved 2021-06-22. <a href="https://matplotlib.org/stable/gallery/scales/symlog_demo.html" target="_blank">https://matplotlib.org/stable/gallery/scales/symlog_demo.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Ainslie, M. A. (2015). A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound. <a href="https://repository.oceanbestpractices.org/bitstream/handle/11329/2340/A-Century-of-Sonar.pdf?sequence=1" target="_blank">https://repository.oceanbestpractices.org/bitstream/handle/11329/2340/A-Century-of-Sonar.pdf?sequence=1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>