Editing Logarithm (section)

===Logarithmic scale===
{{Main|Logarithmic scale}}
[[File:Germany Hyperinflation.svg|A logarithmic chart depicting the value of one [[German gold mark|Goldmark]] in [[German Papiermark|Papiermarks]] during the [[Inflation in the Weimar Republic|German hyperinflation in the 1920s]]|right|thumb|alt=A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale.]]
Scientific quantities are often expressed as logarithms of other quantities, using a ''logarithmic scale''. For example, the [[decibel]] is a [[unit of measurement]] associated with [[logarithmic-scale]] [[level quantity|quantities]]. It is based on the common logarithm of [[ratio]]s—10&nbsp;times the common logarithm of a [[power (physics)|power]] ratio or 20&nbsp;times the common logarithm of a [[voltage]] ratio. It is used to quantify the attenuation or amplification of electrical signals,<ref>{{cite book|contribution=7.5.1 Decibel (dB)|title=Power Quality|first=C.|last=Sankaran|publisher=Taylor & Francis|year=2001|isbn=9780849310409|quote=The decibel is used to express the ratio between two quantities. The quantities may be voltage, current, or power.}}</ref> to describe power levels of sounds in [[acoustics]],<ref>{{Citation|last1=Maling|first1=George C.|editor1-last=Rossing|editor1-first=Thomas D.|title=Springer handbook of acoustics|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-0-387-30446-5|year=2007|chapter=Noise}}, section 23.0.2</ref> and the [[absorbance]] of light in the fields of [[spectrometer|spectrometry]] and [[optics]]. The [[signal-to-noise ratio]] describing the amount of unwanted [[noise (electronic)|noise]] in relation to a (meaningful) [[signal (information theory)|signal]] is also measured in decibels.<ref>{{Citation | last1=Tashev | first1=Ivan Jelev | title=Sound Capture and Processing: Practical Approaches | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-470-31983-3 | year=2009|url={{google books |plainurl=y |id=plll9smnbOIC|page=48}}|page=98}}</ref> In a similar vein, the [[peak signal-to-noise ratio]] is commonly used to assess the quality of sound and [[image compression]] methods using the logarithm.<ref>{{Citation | last1=Chui | first1=C.K. | title=Wavelets: a mathematical tool for signal processing | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | series=SIAM monographs on mathematical modeling and computation | isbn=978-0-89871-384-8 | year=1997|url={{google books |plainurl=y |id=N06Gu433PawC|page=180}}}}</ref>

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the [[moment magnitude scale]] or the [[Richter magnitude scale]]. For example, a 5.0 earthquake releases 32&nbsp;times {{math|(10<sup>1.5</sup>)}} and a 6.0 releases 1000&nbsp;times {{math|(10<sup>3</sup>)}} the energy of a 4.0.<ref>{{Citation|last1=Crauder|first1=Bruce|last2=Evans|first2=Benny|last3=Noell|first3=Alan|title=Functions and Change: A Modeling Approach to College Algebra|publisher=Cengage Learning|location=Boston|edition=4th|isbn=978-0-547-15669-9|year=2008}}, section 4.4.</ref> [[Apparent magnitude]] measures the brightness of stars logarithmically.<ref>{{Citation|last1=Bradt|first1=Hale|title=Astronomy methods: a physical approach to astronomical observations|publisher=[[Cambridge University Press]]|series=Cambridge Planetary Science|isbn=978-0-521-53551-9|year=2004}}, section 8.3, p.&nbsp;231</ref> In [[chemistry]] the negative of the decimal logarithm, the decimal '''{{vanchor|cologarithm}}''', is indicated by the letter p.<ref name="Jens">{{cite journal|author=Nørby, Jens|year=2000|title=The origin and the meaning of the little p in pH|journal=Trends in Biochemical Sciences|volume=25|issue=1|pages=36–37|doi=10.1016/S0968-0004(99)01517-0|pmid=10637613}}</ref> For instance, [[pH]] is the decimal cologarithm of the [[Activity (chemistry)|activity]] of [[hydronium]] ions (the form [[hydrogen]] [[ion]]s {{H+}} take in water).<ref>{{Citation|author=IUPAC|title=Compendium of Chemical Terminology ("Gold Book")|edition=2nd|editor=A. D. McNaught, A. Wilkinson|publisher=Blackwell Scientific Publications| location=Oxford| year=1997| url=http://goldbook.iupac.org/P04524.html|isbn=978-0-9678550-9-7|doi=10.1351/goldbook|author-link=IUPAC|doi-access=free}}</ref> The activity of hydronium ions in neutral water is 10<sup>−7</sup>&nbsp;[[molar concentration|mol·L<sup>−1</sup>]], hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10<sup>4</sup> of the activity, that is, vinegar's hydronium ion activity is about {{math|10<sup>−3</sup> mol·L<sup>−1</sup>}}.

[[Semi-log plot|Semilog]] (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1&nbsp;million to 1&nbsp;trillion to the same space (on the vertical axis) as the increase from 1 to 1&nbsp;million. In such graphs, [[exponential function]]s of the form {{math|1=''f''(''x'') = ''a'' · ''b''{{i sup|''x''}}}} appear as straight lines with [[slope]] equal to the logarithm of {{mvar|b}}. [[Log-log plot|Log-log]] graphs scale both axes logarithmically, which causes functions of the form {{math|1=''f''(''x'') = ''a'' · ''x''{{i sup|''k''}}}} to be depicted as straight lines with slope equal to the exponent&nbsp;{{mvar|k}}. This is applied in visualizing and analyzing [[power law]]s.<ref>{{Citation|last1=Bird|first1=J.O.|title=Newnes engineering mathematics pocket book  |publisher=Newnes|location=Oxford|edition=3rd|isbn=978-0-7506-4992-6|year=2001}}, section 34</ref>