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[[File:Logarithmic Scales.svg|thumb|400px|Various scales: [[Lin-lin graph|lin-lin]], [[Lin-log graph|lin-log]], [[Log-lin graph|log-lin]], and [[Log-log graph|log-log]]. Plotted graphs are: ''y'' = 10<sup> ''x''</sup> (<span style="color:red;">red</span>), ''y'' = ''x'' (<span style="color:green;">green</span>), ''y'' = log<sub>''e''</sub>(''x'') (<span style="color:blue;">blue</span>).]] | |||
A '''logarithmic scale''' is a [[scale (measurement)|scale of measurement]] that displays the value of a [[physical quantity]] using intervals corresponding to orders of magnitude, rather than a standard linear scale. The function of the curve may include an exponent which is what gives it its curved nature. | |||
A simple example is a chart whose vertical or horizontal axis has equally spaced increments that are labeled 1, 10, 100, 1000, instead of 0, 1, 2, 3. Each unit increase on the logarithmic scale thus represents an [[Exponentiation|exponential]] increase in the underlying quantity for the given base (10, in this case). | |||
Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values. The use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size. Some of our [[sense]]s operate in a logarithmic fashion ([[Weber–Fechner law]]), which makes logarithmic scales for these input quantities especially appropriate. In particular our sense of [[hearing (sense)|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 by humans.<ref>{{cite web|url=http://www.sciencedaily.com/releases/2008/05/080529141344.htm|title=Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space | |||
|date=2008-05-30|publisher=ScienceDaily|accessdate=2008-05-31}} <br />'''which references:''' {{cite journal|last=Stanislas|first=Dehaene|coauthors=Véronique Izard, [[Elizabeth Spelke]], and [[Pierre Pica]].|year=2008|title=Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures|journal=Science|volume=320|issue=5880|doi=10.1126/science.1156540|pmid=18511690|pmc=2610411|pages=1217–20}}</ref> | |||
==Definition and base== | |||
Logarithmic scales are either defined for ''ratios'' of the underlying quantity, or one has to agree to measure the quantity in fixed units. Deviating from these units means that the logarithmic measure will change by an ''additive'' constant. The base of the logarithm also has to be specified, unless the scale's value is considered to be a dimensional quantity expressed in generic (indefinite-base) logarithmic units. | |||
==Example scales== | |||
On most logarithmic scales, ''small'' values (or ratios) of the underlying quantity correspond to ''negative'' values of the logarithmic measure. Well-known examples of such scales are: | |||
* [[Richter magnitude scale]] and [[moment magnitude scale]] (MMS) for strength of [[earthquakes]] and [[Motion (physics)|movement]] in the [[earth]]. | |||
* [[Ban (information)|ban and deciban]], for information or weight of evidence; | |||
* [[decibel|bel and decibel]] and [[neper]] for acoustic power (loudness) and electric power; | |||
* [[Cent (music)|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-stop]]s for ratios of [[photographic exposure]]; | |||
* rating low [[probabilities]] by the number of 'nines' in the decimal expansion of the probability of their not happening: for example, a system which will fail with a probability of 10<sup>−5</sup> is 99.999% reliable: "five nines". | |||
* [[Entropy]] in [[thermodynamics]]. | |||
* [[Information]] in [[information theory]]. | |||
* Particle Size Distribution curves of soil | |||
Some logarithmic scales were designed such that ''large'' values (or ratios) of the underlying quantity correspond to ''small'' values of the logarithmic measure. Examples of such scales are: | |||
* [[pH]] for acidity and alkalinity; | |||
* [[Apparent magnitude|stellar magnitude scale]] for brightness of [[star]]s; | |||
* [[Krumbein scale]] for [[Particle size (grain size)|particle size]] in [[geology]]. | |||
* [[Absorbance]] of light by transparent samples. | |||
==Logarithmic units== | |||
Logarithmic units are abstract mathematical units that can be used to express any quantities (physical or mathematical) that are defined on a logarithmic scale, that is, as being proportional to the value of a [[logarithm]] function. In this article, a given logarithmic unit will be denoted using the notation [log ''n''], where ''n'' is a positive real number, and [log ] here denotes the [[indefinite logarithm]] function Log(). | |||
===Examples=== | |||
Examples of logarithmic units include common units of [[information]] and [[entropy]], such as the ''[[bit]]'' [log 2] and the ''[[byte]]'' 8[log 2] = [log 256], also the ''[[nat (information)|nat]]'' [log e] and the ''[[ban (information)|ban]]'' [log 10]; units of relative signal strength magnitude such as the ''[[decibel]]'' 0.1[log 10] and ''bel'' [log 10], ''[[neper]]'' [log e], and other logarithmic-scale units such as the [[Richter scale]] point [log 10] or (more generally) the corresponding order-of-magnitude unit sometimes referred to as a ''factor of ten'' or ''[[Decade (log scale)|decade]]'' (here meaning [log 10], not 10 years). | |||
===Motivation=== | |||
The motivation behind the concept of logarithmic units is that defining a quantity on a logarithmic scale in terms of a logarithm to a specific base amounts to making a (totally arbitrary) choice of a unit of measurement for that quantity, one that corresponds to the specific (and equally arbitrary) logarithm base that was selected. Due to the identity | |||
: <math>\log_b a = \frac{\log_c a}{\log_c b},</math> | |||
the logarithms of any given number ''a'' to two different bases (here ''b'' and ''c'') differ only by the constant factor log<sub>''c''</sub> ''b''. This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity Log(''a'') from one arbitrary unit of measurement (the [log ''c''] unit) to another (the [log ''b''] unit), since | |||
: <math> \mathrm{Log}(a) = (\log_b a)[\log b] = (\log_c a)[\log c]. \, </math> | |||
For example, [[Boltzmann]]'s standard definition of entropy ''S'' = ''k'' ln ''W'' (where ''W'' is the number of ways of arranging a system and ''k'' is [[Boltzmann constant|Boltzmann's constant]]) can also written more simply as just ''S'' = Log(''W''), where "Log" here denotes the [[indefinite logarithm]], and we let ''k'' = [log e]; that is, we identify the physical entropy unit ''k'' with the mathematical unit [log e]. This identity works because | |||
: <math>\ln W = \log_e W = \frac{\mathrm{Log}(W)}{\log e}.</math> | |||
Thus, we can interpret Boltzmann's constant as being simply the expression (in terms of more standard physical units) of the abstract logarithmic unit [log e] that is needed to convert the dimensionless pure-number quantity ln ''W'' (which uses an arbitrary choice of base, namely e) to the more fundamental pure logarithmic quantity Log(''W''), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy. | |||
==Graphic representation==<!-- This section is linked from [[Order of magnitude]] --> | |||
[[File:COB data Tsunami deaths.PNG|right|thumb|400px|A log scale makes it easy to compare values which cover a large range, such as in this map]] | |||
A logarithmic scale is also a graphical scale on one or both sides of a graph where a number ''x'' is printed at a distance ''c''·log(''x'') from the point marked with the number 1. A [[slide rule]] has logarithmic scales, and [[nomogram]]s often employ logarithmic scales. On a logarithmic scale an equal difference in [[order of magnitude]] is represented by an equal distance. The [[geometric mean]] of two numbers is midway between the numbers. | |||
'''Logarithmic [[graph paper]]''', before the advent of computer graphics, was a basic scientific tool. Plots on paper with one log scale can show up [[exponential law]]s, and on log-log paper [[power law]]s, as straight lines (see [[semi-log graph]], [[log-log graph]]). | |||
===Comparing the scales=== | |||
[[File:Comparison of the sequence 1 to 10 and their logs to the base 10.png|thumb|Comparison of the sequence 1 to 10 and their logs to the base 10]]A plot of ''x'' v. log<sub>10</sub>(''x''). Note two things: first, log(''x'') increases quickly at first: by ''x'' = 3, log(''x'') is almost at .5; it is useful to remember that sqrt(10) ~ 3. Second, log(''x'') grows ever more slowly as ''x'' approaches 10; this shows how logarithms can be used to 'tame' large numbers. | |||
=={{anchor|Logarithmic plots}}Logarithmic and semi-logarithmic plots and equations of lines== | |||
Log and semilog scales are best used to view two types of equations (for ease, the natural base 'e' is used): | |||
: <math> (1)\quad Y = \exp(-aX) </math> | |||
: <math> (2)\quad Y = X^b. </math> | |||
In the first case, plotting the equation on a semilog scale (log ''Y'' versus ''X'') gives: log ''Y'' = −''aX'', which is linear.<br /> | |||
In the second case, plotting the equation on a log-log scale (log ''Y'' versus log ''X'') gives: log ''Y'' = ''b'' log ''X'', which is linear.<br /> | |||
When values that span large ranges need to be plotted, a logarithmic scale can provide a means of viewing the data that allows the values to be determined from the graph. '''The logarithmic scale is marked off in distances proportional to the logarithms of the values being represented.''' For example, in the figure below, for both plots, y has the values of: 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. For the plot on the left, the log<sub>10</sub> of the values of ''y'' are plotted on a linear scale. Thus the first value is log<sub>10</sub>(1) = 0; the second value is log<sub>10</sub>(2) = 0.301; the 3rd value is log<sub>10</sub>(3) = 0.4771; the 4th value is log<sub>10</sub>(4) = 0.602, and so on. The plot on the right uses logarithmic (or log, as it is also referred to) scaling on the vertical axis. Note that values where the exponent term is close to an integral fraction of 10 (0.1, 0.2, 0.3, etc.) are shown as 10 raised to the power that yields the original value of y. These are shown for ''y'' = 2, 4, 8, 10, 20, 40, 80 and 100. | |||
{{-}} | |||
[[File:SemiLogPlotDescription.GIF|frame|left|Plots of the log (base 10) of values of ''y'' (see text) on a linear scale (left plot) and of values of ''y'' on a log scale (right plot).]] | |||
{{-}} | |||
Note that for ''y'' = 2 and 20, ''y'' = 10<sup>0.301</sup> and 10<sup>1.301</sup>; for ''y'' = 4 and 40, ''y'' = 10<sup>0.602</sup> and 10<sup>1.602</sup>. This is due to the law that | |||
: <math> \log(AB) = \log(A) + \log(B).\,</math> | |||
So, knowing log<sub>10</sub>(2) = 0.301, the rest can be derived: | |||
: <math> \log_{10} (4) = \log_{10} (2 \times 2) = \log_{10}(2) + \log_{10}(2) = 0.602 </math> | |||
: <math> \log_{10} (20) = \log_{10}(2 \times 10) = \log_{10} (2) + \log_{10}(10) = 1.301. </math> | |||
Note that the values of ''y'' are easily picked off the above figure. By comparison, values of ''y'' less than 10 are difficult to determine from the figure below, where they are plotted on a linear scale, thus confirming the earlier assertion that values spanning large ranges are more easily read from a logarithmically scaled graph. | |||
{{-}} | |||
[[File:LargeSpanLinearPlot.GIF|frame|left|Plot of the values of ''y'' (see text) on a linear scale.]] | |||
{{-}} | |||
===Log-log plots=== | |||
[[File:LogLogPlot of Line.GIF|right|thumb|200px|Plot on log-log scale of equation ''F''(''x'') = (''x''<sup>−10</sup> )(10<sup>20</sup>), which can be expressed as the line: log(''F''(''x'')) = -10 log(''x'') + 20.]] | |||
If both the vertical and horizontal axis of a plot is scaled logarithmically, the plot is referred to as a log-log plot. | |||
{{main|log-log graph}} | |||
===Semi logarithmic plots=== | |||
If only the [[ordinate]] or [[abscissa]] is scaled logarithmically, the plot is referred to as a semi logarithmic plot. | |||
{{main|semi-log graph}} | |||
==Estimating values in a diagram with logarithmic scale== | |||
One method for accurate determination of values on a logarithmic axis is as follows: | |||
# Measure the distance from the point on the scale to the closest decade line with lower value with a ruler. | |||
# Divide this distance by the length of a decade (the length between two decade lines). | |||
# The value of your chosen point is now the value of the nearest decade line with lower value times 10<sup>''a''</sup> where ''a'' is the value found in step 2. | |||
Example: What is the value that lies halfway between the 10 and 100 decades on a logarithmic axis? Since it is the halfway point that is of interest, the quotient of steps 1 and 2 is 0.5. The nearest decade line with lower value is 10, so the halfway point's value is (10<sup>0.5</sup>) × 10 = 10<sup>1.5</sup> ≈ 31.62. | |||
To estimate where a value lies within a decade on a logarithmic axis, use the following method: | |||
# Measure the distance between consecutive decades with a ruler. You can use any units provided that you are consistent. | |||
# Take the log (value of interest/nearest lower value decade) multiplied by the number determined in step one. | |||
# Using the same units as in step 1, count as many units as resulted from step 2, starting at the lower decade. | |||
Example: To determine where 17 is located on a logarithmic axis, first use a ruler to measure the distance between 10 and 100. If the measurement is 30mm on a ruler (it can vary — ensure that the same scale is used throughout the rest of the process). | |||
: [log (17/10)] × 30 = 6.9 | |||
''x'' = 17 is then 6.9mm after ''x'' = 10 (along the ''x''-axis). | |||
===Logarithmic interpolation=== | |||
Interpolating logarithmic values is very similar to interpolating linear values. In linear interpolation, values are determined through equal ratios. For example, in linear interpolation, a line that increases one ordinate (y-value) for every two abscissa (x-value) has a ratio (also known as slope or rise-over-run) of 1/2. To determine the ordinate or abscissa of a particular point, you must know the other value. The calculation of the ordinate corresponding to an abscissa of 12 in the example below is as follows: | |||
: 1/2 = ''Y''/12 | |||
''Y'' is the unknown ordinate. Using cross-multiplication, ''Y'' can be calculated and is equal to 6. | |||
In logarithmic interpolation, a ratio of logarithmic values is set equal to a ratio of linear values. For example, consider a log base 10 scale graph of paper reams sold per day measuring 19{{fraction|1|32}} inches from 1 to 10. How many reams were sold in a day if the value on the graph is 11{{fraction|1|32}} between 1 and 10? To solve this problem, it is necessary to use a basic logarithmic definition: | |||
: log(''A'') − log(''B'') = log(''A''/''B'') | |||
Decade lines, those values that denote powers of the log base, are also important in logarithmic interpolation. Locate the lower decade line. It is the closest decade line to the number you are evaluating that is lower than that number. Decade lines begin at 1. The next decade line is the first power of your log base. For log base 10, the first decade line is 1, the second is 10, the third is 100, and so on. | |||
The ratio of linear values is the number of units from the lower decade line to the value of interest (11{{fraction|1|32}} in this example, since the lower decade line in this example is 1) divided by the total number of units between the lower decade line and the upper decade line (the upper decade line is 10 in this example). Therefore, the linear ratio is: | |||
:11/19 | |||
Notice that the units (1/32 inch) are removed from the equation because both measurements are in the same units. Conversion to a single unit before calculating the ratio is required if the measurements were made in different units. | |||
The logarithmic ratio uses the same graphical measurements as the linear ratio. The difference between the log of the upper decade line (10) and the log of the lower decade line (1) represents the same graphical distance as the total number of units between the two decade lines in the linear ratio (19{{fraction|1|32}}nds of an inch). Therefore, the lower part of the logarithmic ratio (the bottom part of the fraction) is: | |||
:log(10) − log(1) | |||
The upper part of the logarithmic ratio (the top part of the fraction) represents the same graphical distance as the number of units between the value of interest (number of reams of paper sold) and the lower decade line in linear ratio (11{{fraction|1|32}}nds of an inch). The unknown in this ratio is the value of interest, which we will call ''X''. Therefore, the top part of the fraction is: | |||
:log(''X'') − log(1) | |||
The logarithmic ratio is: | |||
:[log(X) − log(1)]/[log(10) − log(1)] | |||
The linear ratio is equal to the logarithmic ratio. Therefore, the equation required to determine the number of paper reams sold in a particular day is: | |||
:11/19 = [log(X) − log(1)]/[log(10) − log(1)] | |||
This equation can be rewritten using the logarithmic definition mentioned above: | |||
:11/19 = log(X/1)/log(10) | |||
log(10) = 1, therefore: | |||
:11/19 = log(''X''/1) | |||
To remove the "log" from the right side of the equation, both sides must be used as exponents for the number 10, meaning 10 to the power of 11/19 and 10 to the power of log(''X''/1). The "log" function and the "10 to the power of" function are reciprocal and cancel each other out, leaving: | |||
:10<sup>11/19</sup> = ''X''/1 | |||
Now both sides must be multiplied by 1. While the 1 drops out of this equation, it is important to note that the number ''X'' is divided by is the value of the lower decade line. If this example involved values between 10 and 100, the equation would include ''X''/10 instead of ''X''/1. | |||
:10<sup>11/19</sup> = ''X'' | |||
''X'' = 3.793 reams of paper. | |||
== See also == | |||
{{Portal|Mathematics}} | |||
* [[Entropy]] | |||
* [[Logarithm]] | |||
* [[Preferred number]] | |||
===Units of information=== | |||
* [[ban (information)|ban]] [log 10] | |||
* [[bit]] [log 2] | |||
* [[byte]] 8[log 2] = [log 256] | |||
* [[nat (information)|nat]] [log e] | |||
===Units of relative signal strength=== | |||
* [[bel]] [log 10] | |||
* [[decibel]] 0.1[log 10] | |||
* [[neper]] [log e] | |||
===Scale=== | |||
* [[Decade (log scale)|Decade]] | |||
* [[Order of magnitude]] | |||
===Applications=== | |||
* [[optical density]] [log 10] | |||
==References== | |||
{{Reflist}} | |||
==External links== | |||
* {{en icon}} [http://legeneraliste.perso.sfr.fr/?p=echelle_log_eng Why using logarithmic scale to display share prices?] | |||
* {{Commonscat-inline}} | |||
* [http://www.peregraph.com/documents/graphpaper Example Logarithmic Graph Paper Template] | |||
{{DEFAULTSORT:Logarithmic Scale}} | |||
[[Category:Logarithmic scales of measurement]] | |||
Revision as of 22:14, 14 January 2014
A logarithmic scale is a scale of measurement that displays the value of a physical quantity using intervals corresponding to orders of magnitude, rather than a standard linear scale. The function of the curve may include an exponent which is what gives it its curved nature.
A simple example is a chart whose vertical or horizontal axis has equally spaced increments that are labeled 1, 10, 100, 1000, instead of 0, 1, 2, 3. Each unit increase on the logarithmic scale thus represents an exponential increase in the underlying quantity for the given base (10, in this case).
Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values. The use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size. 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 by humans.[1]
Definition and base
Logarithmic scales are either defined for ratios of the underlying quantity, or one has to agree to measure the quantity in fixed units. Deviating from these units means that the logarithmic measure will change by an additive constant. The base of the logarithm also has to be specified, unless the scale's value is considered to be a dimensional quantity expressed in generic (indefinite-base) logarithmic units.
Example scales
On most logarithmic scales, small values (or ratios) of the underlying quantity correspond to negative values of the logarithmic measure. Well-known examples of such scales are:
- Richter magnitude scale and moment magnitude scale (MMS) for strength of earthquakes and movement in the earth.
- ban and deciban, for information or weight of evidence;
- bel and decibel and neper for acoustic power (loudness) and electric power;
- 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;
- rating low probabilities by the number of 'nines' in the decimal expansion of the probability of their not happening: for example, a system which will fail with a probability of 10−5 is 99.999% reliable: "five nines".
- Entropy in thermodynamics.
- Information in information theory.
- Particle Size Distribution curves of soil
Some logarithmic scales were designed such that large values (or ratios) of the underlying quantity correspond to small values of the logarithmic measure. Examples of such scales are:
- pH for acidity and alkalinity;
- stellar magnitude scale for brightness of stars;
- Krumbein scale for particle size in geology.
- Absorbance of light by transparent samples.
Logarithmic units
Logarithmic units are abstract mathematical units that can be used to express any quantities (physical or mathematical) that are defined on a logarithmic scale, that is, as being proportional to the value of a logarithm function. In this article, a given logarithmic unit will be denoted using the notation [log n], where n is a positive real number, and [log ] here denotes the indefinite logarithm function Log().
Examples
Examples of logarithmic units include common units of information and entropy, such as the bit [log 2] and the byte 8[log 2] = [log 256], also the nat [log e] and the ban [log 10]; units of relative signal strength magnitude such as the decibel 0.1[log 10] and bel [log 10], neper [log e], and other logarithmic-scale units such as the Richter scale point [log 10] or (more generally) the corresponding order-of-magnitude unit sometimes referred to as a factor of ten or decade (here meaning [log 10], not 10 years).
Motivation
The motivation behind the concept of logarithmic units is that defining a quantity on a logarithmic scale in terms of a logarithm to a specific base amounts to making a (totally arbitrary) choice of a unit of measurement for that quantity, one that corresponds to the specific (and equally arbitrary) logarithm base that was selected. Due to the identity
the logarithms of any given number a to two different bases (here b and c) differ only by the constant factor logc b. This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity Log(a) from one arbitrary unit of measurement (the [log c] unit) to another (the [log b] unit), since
![{\displaystyle \mathrm {Log} (a)=(\log _{b}a)[\log b]=(\log _{c}a)[\log c].\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/003fed38c16c8d570ff548e21e1836bdd6bdcdac)
For example, Boltzmann's standard definition of entropy S = k ln W (where W is the number of ways of arranging a system and k is Boltzmann's constant) can also written more simply as just S = Log(W), where "Log" here denotes the indefinite logarithm, and we let k = [log e]; that is, we identify the physical entropy unit k with the mathematical unit [log e]. This identity works because
Thus, we can interpret Boltzmann's constant as being simply the expression (in terms of more standard physical units) of the abstract logarithmic unit [log e] that is needed to convert the dimensionless pure-number quantity ln W (which uses an arbitrary choice of base, namely e) to the more fundamental pure logarithmic quantity Log(W), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.
Graphic representation
A logarithmic scale is also a graphical scale on one or both sides of a graph where a number x is printed at a distance c·log(x) from the point marked with the number 1. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. On a logarithmic scale an equal difference in order of magnitude is represented by an equal distance. The geometric mean of two numbers is midway between the numbers.
Logarithmic graph paper, before the advent of computer graphics, was a basic scientific tool. Plots on paper with one log scale can show up exponential laws, and on log-log paper power laws, as straight lines (see semi-log graph, log-log graph).
Comparing the scales
A plot of x v. log10(x). Note two things: first, log(x) increases quickly at first: by x = 3, log(x) is almost at .5; it is useful to remember that sqrt(10) ~ 3. Second, log(x) grows ever more slowly as x approaches 10; this shows how logarithms can be used to 'tame' large numbers.
<Logarithmic plots>...</Logarithmic plots>Logarithmic and semi-logarithmic plots and equations of lines
Log and semilog scales are best used to view two types of equations (for ease, the natural base 'e' is used):
In the first case, plotting the equation on a semilog scale (log Y versus X) gives: log Y = −aX, which is linear.
In the second case, plotting the equation on a log-log scale (log Y versus log X) gives: log Y = b log X, which is linear.
When values that span large ranges need to be plotted, a logarithmic scale can provide a means of viewing the data that allows the values to be determined from the graph. The logarithmic scale is marked off in distances proportional to the logarithms of the values being represented. For example, in the figure below, for both plots, y has the values of: 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. For the plot on the left, the log10 of the values of y are plotted on a linear scale. Thus the first value is log10(1) = 0; the second value is log10(2) = 0.301; the 3rd value is log10(3) = 0.4771; the 4th value is log10(4) = 0.602, and so on. The plot on the right uses logarithmic (or log, as it is also referred to) scaling on the vertical axis. Note that values where the exponent term is close to an integral fraction of 10 (0.1, 0.2, 0.3, etc.) are shown as 10 raised to the power that yields the original value of y. These are shown for y = 2, 4, 8, 10, 20, 40, 80 and 100.
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FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE
To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.
Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.
The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.
As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com
Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback
Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris
FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE
To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.
Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.
The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.
As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com
Note that for y = 2 and 20, y = 100.301 and 101.301; for y = 4 and 40, y = 100.602 and 101.602. This is due to the law that
So, knowing log10(2) = 0.301, the rest can be derived:
Note that the values of y are easily picked off the above figure. By comparison, values of y less than 10 are difficult to determine from the figure below, where they are plotted on a linear scale, thus confirming the earlier assertion that values spanning large ranges are more easily read from a logarithmically scaled graph.
Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback
Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris
FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE
To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.
Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.
The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.
As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com
Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback
Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris
FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE
To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.
Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.
The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.
As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com
Log-log plots
If both the vertical and horizontal axis of a plot is scaled logarithmically, the plot is referred to as a log-log plot. Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
Semi logarithmic plots
If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi logarithmic plot.
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
Estimating values in a diagram with logarithmic scale
One method for accurate determination of values on a logarithmic axis is as follows:
- Measure the distance from the point on the scale to the closest decade line with lower value with a ruler.
- Divide this distance by the length of a decade (the length between two decade lines).
- The value of your chosen point is now the value of the nearest decade line with lower value times 10a where a is the value found in step 2.
Example: What is the value that lies halfway between the 10 and 100 decades on a logarithmic axis? Since it is the halfway point that is of interest, the quotient of steps 1 and 2 is 0.5. The nearest decade line with lower value is 10, so the halfway point's value is (100.5) × 10 = 101.5 ≈ 31.62.
To estimate where a value lies within a decade on a logarithmic axis, use the following method:
- Measure the distance between consecutive decades with a ruler. You can use any units provided that you are consistent.
- Take the log (value of interest/nearest lower value decade) multiplied by the number determined in step one.
- Using the same units as in step 1, count as many units as resulted from step 2, starting at the lower decade.
Example: To determine where 17 is located on a logarithmic axis, first use a ruler to measure the distance between 10 and 100. If the measurement is 30mm on a ruler (it can vary — ensure that the same scale is used throughout the rest of the process).
- [log (17/10)] × 30 = 6.9
x = 17 is then 6.9mm after x = 10 (along the x-axis).
Logarithmic interpolation
Interpolating logarithmic values is very similar to interpolating linear values. In linear interpolation, values are determined through equal ratios. For example, in linear interpolation, a line that increases one ordinate (y-value) for every two abscissa (x-value) has a ratio (also known as slope or rise-over-run) of 1/2. To determine the ordinate or abscissa of a particular point, you must know the other value. The calculation of the ordinate corresponding to an abscissa of 12 in the example below is as follows:
- 1/2 = Y/12
Y is the unknown ordinate. Using cross-multiplication, Y can be calculated and is equal to 6.
In logarithmic interpolation, a ratio of logarithmic values is set equal to a ratio of linear values. For example, consider a log base 10 scale graph of paper reams sold per day measuring 19Template:Fraction inches from 1 to 10. How many reams were sold in a day if the value on the graph is 11Template:Fraction between 1 and 10? To solve this problem, it is necessary to use a basic logarithmic definition:
- log(A) − log(B) = log(A/B)
Decade lines, those values that denote powers of the log base, are also important in logarithmic interpolation. Locate the lower decade line. It is the closest decade line to the number you are evaluating that is lower than that number. Decade lines begin at 1. The next decade line is the first power of your log base. For log base 10, the first decade line is 1, the second is 10, the third is 100, and so on.
The ratio of linear values is the number of units from the lower decade line to the value of interest (11Template:Fraction in this example, since the lower decade line in this example is 1) divided by the total number of units between the lower decade line and the upper decade line (the upper decade line is 10 in this example). Therefore, the linear ratio is:
- 11/19
Notice that the units (1/32 inch) are removed from the equation because both measurements are in the same units. Conversion to a single unit before calculating the ratio is required if the measurements were made in different units.
The logarithmic ratio uses the same graphical measurements as the linear ratio. The difference between the log of the upper decade line (10) and the log of the lower decade line (1) represents the same graphical distance as the total number of units between the two decade lines in the linear ratio (19Template:Fractionnds of an inch). Therefore, the lower part of the logarithmic ratio (the bottom part of the fraction) is:
- log(10) − log(1)
The upper part of the logarithmic ratio (the top part of the fraction) represents the same graphical distance as the number of units between the value of interest (number of reams of paper sold) and the lower decade line in linear ratio (11Template:Fractionnds of an inch). The unknown in this ratio is the value of interest, which we will call X. Therefore, the top part of the fraction is:
- log(X) − log(1)
The logarithmic ratio is:
- [log(X) − log(1)]/[log(10) − log(1)]
The linear ratio is equal to the logarithmic ratio. Therefore, the equation required to determine the number of paper reams sold in a particular day is:
- 11/19 = [log(X) − log(1)]/[log(10) − log(1)]
This equation can be rewritten using the logarithmic definition mentioned above:
- 11/19 = log(X/1)/log(10)
log(10) = 1, therefore:
- 11/19 = log(X/1)
To remove the "log" from the right side of the equation, both sides must be used as exponents for the number 10, meaning 10 to the power of 11/19 and 10 to the power of log(X/1). The "log" function and the "10 to the power of" function are reciprocal and cancel each other out, leaving:
- 1011/19 = X/1
Now both sides must be multiplied by 1. While the 1 drops out of this equation, it is important to note that the number X is divided by is the value of the lower decade line. If this example involved values between 10 and 100, the equation would include X/10 instead of X/1.
- 1011/19 = X
X = 3.793 reams of paper.
See also
Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.
Units of information
Units of relative signal strength
Scale
Applications
- optical density [log 10]
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
External links
- Nurse Manager Wesley Kamm from North York, usually spends time with hobbies and interests such as dogs, property developers in condo new launch singapore and baking. Finds the planet an motivating place having spent 9 weeks at City of Graz – Historic Centre and Schloss Eggenberg. Why using logarithmic scale to display share prices?
- Template:Commonscat-inline
- Example Logarithmic Graph Paper Template
- ↑ Template:Cite web
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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