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Logarithmic scale
Definition and basic
Logarithmic scales are either defined for ratios of the base amount, or you have to be according to measure the quantity in fixed units. Deviating from these units means that the logarithmic measure is changed by an additive constant. The base of logarithms is also to be specified, unless the value of the scale are considered a dimensional quantity expressed in generic (indefinite-base) logarithmic units.
Example scales
In most logarithmic scales, small values (or reasons) for the underlying quantity correspond to negative values of the logarithmic measure. well known examples of such scales are:
Richter scale and moment magnitude scale of the scale (MMS) for strength of earthquakes and earth movement.
prohibition and decibels, information or weight of the evidence;
neper bel and decibel and sound power (volume) and electricity;
percent, lower than second, second largest, and the eighth for the relative pitch of the notes of music;
logit odds in statistics;
Scale Palermo Technical Impact Hazard;
Logarithmic timeline;
diaphragms counting relations exposure photo;
Low probability rating by the number of 'nines' in the decimal expansion of the probability that is not happening: For example, a system that will fail with a probability of 10-5 is 99.999% reliable: "five nines."
The entropy in thermodynamics.
The information in information theory.
Particle size distribution curves of soil
Some scales log is designed so that large values (or reasons) for the underlying quantity correspond to small values of the logarithmic measure. Examples of such scales are:
acid pH;
stellar magnitude scale for brightness of stars;
Krumbein scale for particle size in geology.
Absorbance of light by samples of transparency.
Log units
logarithmic units are abstract units mathematics that can be used to express the quantities (physical or mathematical) that are defined on a logarithmic scale, that is, as it is proportional to a logarithmic function. In this article, a given log unit denoted using the notation [log n], where n is a positive real number, and [register] here denotes the indefinite logarithm log ().
Examples
Logarithmic units are examples of common data units and [2 log] entropy as the bit and byte 8 [log 2] = [log 256], also [log e] nat and prohibition [log 10]; units relative magnitude of the signal intensity as the decibel 0.1 [log 10] and [log 10] bel [log e] neper, logarithmic scale and other units, such as point Richter scale [log 10] or (in general) the corresponding order magnitude unit sometimes referred to as a factor of ten or ten years (in this sense [log 10], 10 years).
Motivation
The motivation behind of the concept of logarithmic units is that the definition of a quantity on a logarithmic scale in terms of a logarithm to base specific amounts to choice (completely arbitrary) of a unit of measurement of this quantity, which corresponds to the logarithm specific (and equally arbitrary) that was selected. Because identity
,
the logarithms of any number given to two bases different (in this case B and C) differ only by the constant factor logC b. This constant can be considered to represent the conversion factor for converting a digital representation of the pure (indefinite) logarithmic quantity Log (a) an arbitrary unit of measurement ([log c] of the unit) to another (the [log b] unit), as
For example, the standard definition of Boltzmann entropy S = k ln W (where W is the number of forms of organization of a system and k is the Boltzmann constant), can also be written more simply as just S = log (W) "Sign" here denotes the indefinite logarithm, and is k = [log t], ie k identifies the physical unit to unit entropy mathematics [log e]. This works because the identity
.
Therefore, we interpret the Boltzmann constant to be simply the [deleted] in terms of more standard physical units) of the abstract-log [log t] that is needed to convert the sheer size of the series in number of W (which uses an arbitrary choice of the base, namely, e) to the most fundamental pure logarithmic number of records (W), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.
Graphing
A log scale makes easy to compare values covering a wide range, as in this map
A logarithmic scale is also a graphic scale in one or both sides of a graph where a number x is printed on an obstruction at a distance (x) from the point marked with number 1. A slide rule has logarithmic scales, and nomograms often logarithmic scales used. On a logarithmic scale an equal difference in order of magnitude is represented by the same distance. The geometric mean of two numbers is halfway between the numbers.
logarithmic graph paper, before the advent of computer graphics, was a basic scientific tool. Plots logarithmic paper can demonstrate to the exponential laws, and the law journal paper feed, as straight lines (see figure semi-log, logarithmic graph).
Logarithmic and semi-logarithmic plots and equations of lines
Registration and semilogarithmic scales are best used to see two types of equations (for ease, 'and' natural base used):
Y = exp (aX)
Y = Xb
In the first case, the layout of the equation in a semi-logarithmic scale (log Y versus X) gives: log Y = X, which is linear.
In the second case, the layout of the equation in a log-log scale (log Y versus log X) gives: log Y = log b X, which is linear.
When the values that span large ranges, it is necessary to draw a logarithmic scale can provide a data display that allows values to be determined on the graph. The logarithmic scale is marked at distances proportional to logarithms of the values it represents. For example, in the following figure, both plots, and has values: 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. On the left plot, the log 10 values and plotted on a linear scale. So the first value is log10 (1) = 0, the second value is log10 (2) = 0.301, the third is the value log10 (3) = 0.4771, the value 4 is log10 (4) = 0.602, and so on. The plot on the right uses logarithmic (Or registration, as it also known) of scale on the vertical axis. Note that the maximum values in the exponent is about a member of the fraction of 10 (0.1, 0.2, 0.3, etc) are shown as 10 to the power produced by the original value of y. These are shown for y = 2, 4, 8, 10, 20, 40, 80 and 100.
Plots the log (base 10) values of y (see text) on a linear scale (left picture) and the y values on a logarithmic scale (right plot).
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 because the law
So, knowing log10 (2) = 0.301, the rest can be derived:
Note that the values of and are easily picked off the previous figure. In comparison, values and less 10 are difficult to determine from the figure below, which are represented on a linear scale, thus confirming the earlier claim that the values ranging large ranges are easier to read on a logarithmic scale graph.
Plot of the values of y (see text) in a linear scale.
Login log plots
Plot in log-log scale of the equation F (x) = (x10) (1020), which can be expressed as the line: log (F (x)) = -10 Log (x) + 20.
If both the vertical and horizontal axis of a logarithmic plot, the plot is known as a log-log plot. Equation a line in a log-log scale would be:
log10 (F (x)) = mlog10 (x) + b,
F (x) = (XM) (10b)
where m is the slope and b is the point of intersection with the log frame.
Slope of a log-log plot
Find the slope of a log-log plot with ratios
To find the slope of the plot, two selected points on the x axis, such as X1 and X2. Using the above equation:
and
The slope m is found by taking the difference:
where F1 is short for F (x1) and similarly for F2. The right figure shows the formula. Note that the pending in the example of the figure is negative. The formula also provides a negative slope, as shown in the following property of logarithms:
Find function of the log-log plot
The above procedure is now reversed to find the shape of the function F (x) with its (assumed) known log-log plot. To find the function F, choose a fixed point (x0, F0), where F0 is the abbreviation for F (x 0), somewhere on the straight line in the graph above and also another arbitrary point (x 1, F1) in the same graph. Then, over the slope formula:
leading to
Note that 10log10 (F1) = F1. Therefore, records can be inverted to find:
or
,
which means that
In other words, F is proportional axa the power of the slope of the line of logarithmic graph. In particular, a straight line on a log-log plot containing points (F0, x0) and (F1, x1) will function:
,
Of course, the opposite is also true: any function of the form
will have a straight line the log-log plot, where the slope of the line.
Semi logarithmic plots
If only the ordinate or abscissa is logarithmic scale The plot concerns a semi logarithmic plot. The equation of a line with a y-axis logarithmic scale would be:
log10 (F (x)) = mx + B
F (x) = 10 (x + b) = (10M) (10 b)
The equation of a line in a plot where the x-axis logarithmic scale would
F (x) = mlog10 (x) + b.
Estimated values in a diagram with logarithmic scale
A method for accurate determination of values in a logarithmic axis is:
Measure the distance from the point of the scale to the nearest decade line with a value lower with a ruler.
Divide this distance by the length of a decade (the distance between two lines decade).
The value of your chosen point is now worth more next decade line with lower value 10 a time where is the value found in step 2.
Example: What is the value that is halfway between 10 and 100 years on a logarithmic axis? Since it is the middle point that is of interest, the ratio of steps 1 and 2 is 0.5. Line next decade with a lower value is 10, so that the value of the midpoint is (100.5) 10 = 101.5 31.62.
For estimated value is 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 they are compatible.
Take the log (value of the interest / decade near the lower) multiplied by the number determined in step one.
Using the same units as in step 1, count as many units as a result of step 2 from the lower decade.
Example: determine where 17 is a logarithmic axis, you first use a ruler to measure the distance between 10 and 100. If the measurement is 30 mm on a ruler (which may vary ensure that the same scale used in the rest of the process).
[Log (17/10)] 30 = 6.9
x = 17 then after 6.9mm x = 10 (along x axis).
Logarithmic interpolation
Tween log values is very similar to linear interpolation of values. In the interpolation linear, the values are determined by relations of equality. For example, linear interpolation, a line that increases in an orderly (and value) for each axis abscissa two (value x) has a relationship (also known as the slope or the rise-over term) 1 / 2. To determine the ordinate or abscissa of a point, must know the value of others. The calculation of the ordinate corresponding to an abscissa of 12 in this example is:
1 / 2 = Y/12
And it is the orderly unknown. Use cross-multiplication, and can be calculated and is equal to 6.
In the logarithmic interpolation, a list of log values is equal to a ratio of linear values. For example, consider a base level of graphic record of 10 reams of paper sold per day is 19 1/32nds of an inch from 1 to 10. How reams sold in one day if the value of the figure is 11 1/32nds between 1 and 10? To resolve this problem, it is necessary to use a logarithmic base definition:
log (A) – log (B) = log (A / B)
Ten lines, the values denote base skills registration are also important in logarithmic interpolation. Find the line decade low. Is the line closest to the decade the number who are evaluating which is lower than that number. Decade lines begin at 1. The line of the next decade is the first power of its registration database. To the base register 10, the line of the first decade is 1, the second is 10, the third is 100, and so on.
The ratio of linear values is the number of line units decade lower than the value of interest (11 1/32nds in this example, since the decade bottom line in this example is 1) divided by the total number of units to line and ten years younger than decade-line (the top line of ten years is 10 in this example). Therefore, linear relationship:
11/19
Notice that the units (1/32nds of an inch) was removed from the equation, because both are in the same units. The conversion to a single unit before calculating the proportion is necessary if the measurements were made in different units.
The logarithmic relationship used the same measures as the linear plot. The difference between the registration of the top decade online (10) and the registration of the lower decade line (1) represents the same distance graph the total number of units between the two lines decade in the linear relationship (19 1/32nds of an inch). Therefore, the bottom of the logarithmic relationship (the bottom of the fraction) is:
log (10) – log (1)
The upper part of the relationship logarithmic (upper fraction) represents the same distance graph the number of units between the value of interest (number of reams of paper sold) and the line decade low linear relationship (11 1/32nds of an inch). The stranger in this regard is the value of the interest, which we define as "X". Therefore, the top of the fraction is:
log (X) – Log (1)
The relationship is logarithmic:
[Log (X) - log (1)] / [Log (10) - log (1)]
The linear relationship is equal to the logarithmic relationship. Therefore, the equation to determine the number of reams of Paper is sold in a particular day is:
11/19 = [log (X) - log (1)] / [log (10) - Log (1)]
This equation can be rewritten log using the definition above:
11/19 = log (X / 1) / log (10)
log (10) = 1, therefore:
11/19 = Log (X / 1)
In order to eliminate the "record" on the right side of the equation, both sides must be used as exponents for the number 10, ie 10 to the power of 11/19 and 10 to the power of log (X / 1). The "record" and the role of "10 to the power of the" function are reciprocal and cancel the other, leaving:
10 ^ (11/19) = X / 1
Now both sides must be multiplied by 1. While one is out of this equation, is important to note that the number X is divided by the value of the line ten years younger. If this example was to between 10 and 100, the equation includes "X/10" instead of "X / 1."
10 ^ (11/19) = X
X = 3,793 reams of paper.
References
^ "Slide rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping About the Area Number." ScienceDaily. 30.5.2008. Http: / / www.sciencedaily.com/releases/2008/05/080529141344.htm. Retrieved on 2008-05-31.
references to: Stanislas Dehaene, Izard Vronique, Elizabeth Spelke and Pierre Pica. (2008). "Log or Linear? Intuition different numerical scale in the Amazonian Indian and Western cultures. "Science 320 (5880): 1217. doi: 10.1126/science.1156540. PMID 18511690.
See also
Mathematics portal
Preferred number
Logarithm
Indefinite logarithm
Entropy
Information units
bit [Log 2]
byte 8 [log 2] = [log 256]
nat [log e]
ban [Log 10]
Units of relative signal strength
bel [log 10]
decibel 0.1 [log 10]
neper [log e]
Scale
Order of magnitude
Decade
Applications
Eighth
pH
Richter scale
External Links
The media logarithmic scale related to Wikipedia, the free encyclopedia
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