Various scales: lin-lin, lin-log, log-lin and log-log. In Science and Engineering, a semi-log graph or semi-log plot is a way of visualizing data that are changing with an exponential relationship In Science and Engineering, a semi-log graph or semi-log plot is a way of visualizing data that are changing with an exponential relationship In Science and Engineering, a log-log graph or log-log plot is a two-dimensional graph of numerical data that uses Logarithmic scales on both Plotted graphs are: y = x (green), y = 10x (red), y = log(x) (blue).

A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. The concept of scale is applicable if a system is represented proportionally by another system In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce A physical Quantity is a physical property that can be quantified

Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range. Some of our senses operate in a logarithmic fashion (Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. Senses are the physiological methods of Perception. The senses and their operation classification and theory are overlapping topics studied by a variety of fields The Weber–Fechner law attempts to describe the relationship between the physical magnitudes of stimuli and the perceived intensity of the stimuli In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children and 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. The Richter magnitude scale, or more correctly local magnitude M L scale assigns a single number to quantify the amount of seismic energy released TalkMoment magnitude scale#Real world examples please.--> The moment magnitude scale An earthquake is the result of a sudden release of energy in the Earth 's crust that creates Seismic waves Earthquakes are recorded with a Seismometer In Physics, motion means a constant change in the location of a body EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001
• 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. A ban, sometimes called a hartley (symbol Hart) or a dit (abbreviation of d ecimal dig' it') is a Logarithmic unit which In Statistics, the use of Bayes factors is a Bayesian alternative to classical Hypothesis testing. The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to For Neper as a mythological god see Neper (mythology, for the lunar crater named Neper see Neper (crater, and for the Scottish mathematician phycisist and The cent is a logarithmic unit of measure used for musical intervals. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the A major second () also called a whole step or a whole tone, is a Musical interval that occurs between the first and second degrees of a In Music, an octave ( is the the use of which is "common in most musical systems Music is an Art form in which the medium is Sound organized in Time. The logit function is an important part of Logistic regression: for more information please see that article In Probability theory and Statistics the odds in favour of an event or a Proposition are the quantity p  / (1 &minus  Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. The Palermo Technical Impact Hazard Scale is a Logarithmic scale used by Astronomers to rate the potential hazard of impact of a near-earth object (NEO A logarithmic timeline is a Timeline laid out according to a Logarithmic scale. In Photography, exposure is the total amount of Light allowed to fall on the photographic medium ( Photographic film or Image sensor) during the Probability is the likelihood or chance that something is the case or will happen 999% reliable: "five nines".
• Entropy in thermodynamics. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning "
• Information in information theory. Information as a concept has a diversity of meanings from everyday usage to technical settings Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information.
• 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;
• stellar magnitude scale for brightness of stars;
• Krumbein scale for particle size in geology. pH is the measure of the acidity or alkalinity of a Solution. The apparent magnitude ( m) of a celestial body is a measure of its Brightness as seen by an observer on Earth, normalized to the value A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth Geology (from Greek γη gê, "earth" and λόγος Logos, "speech" lit
• Absorbance of light by transparent samples. In Spectroscopy, the absorbance A is defined as A_\lambda = -\log_{10}(I/I_0\ where I is the intensity of light

Graphic representation

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 nomograms often employ logarithmic scales. The slide rule, also known as a slipstick, is a mechanical Analog computer. This article is about the graphical devices called nomograms For a description of the Japanese puzzle see Nonogram. On a logarithmic scale an equal difference in order of magnitude is represented by an equal distance. An order of magnitude is the class of scale or magnitude of any amount where each class contains values of a fixed ratio to the class preceding it The geometric mean of two numbers is midway between the numbers. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers

Logarithmic graph paper, before the advent of computer graphics, was a basic scientific tool. Format and availability of graph paper Graph paper is available either as Loose leaf paper or bound in Notebooks It is becoming Plots on paper with one log scale can show up exponential laws, and on log-log paper power laws, as straight lines (see semilog graph, log-log graph). Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A power law is any Polynomial relationship that exhibits the property of Scale invariance. In Science and Engineering, a semi-log graph or semi-log plot is a way of visualizing data that are changing with an exponential relationship In Science and Engineering, a log-log graph or log-log plot is a two-dimensional graph of numerical data that uses Logarithmic scales on both

Logarithmic and semi-logarithmic plots and equations of lines

Basicaly, log & semilog scales are best used to view two types of equations (for ease, the natural base 'e' is used):
(i) Y=exp(-aX)
(ii) Y=Xb
In the first case, ploting the equation on a semilog scale (logY vs. X) gives: logY=-aX, which is linear.
In the second case, ploting the equation on a log-log scale (logY vs. logX) gives: logY=b*logX, 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.

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 = 100. 301 and 101. 301; for y = 4 and 40, y = 100. 602 and 101. 602. This is due to the law that

$\log(AB) = \log(A) + \log(B).\,$

So, knowing log10(2) = 0. 301, the rest can be derived:

log10(4) = log10(2 × 2) = log10(2) + log10(2) = 0. 602
log10(20) = log10(2 × 10) = log10(2) + log10(10) = 1. 301

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.

Plot of the values of y (see text) on a linear scale.

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. The equation for a line on a log-log scale would be:

log(F(x)) = m log(x) + b,
F(x) = (xm)(10b),

where m is the slope and b is the intercept point on the log plot. The example plot shown below is for the equation log(F(x)) = m log(x) + b, for m = −10, b = 20.

Plot on log-log scale of equation F(x) = (x−10 )(1020).

Slope of a log-log plot
Finding the slope of a log-log plot using ratios

To find the slope of the plot, two points are selected on the x-axis, say x1 and x2. Using the above equation:

$\mathrm {log}[F (x_1)] = m\ \mathrm{log}(x_1) + b \ ,$

and

$\mathrm {log}[F (x_2)] = m\ \mathrm{log}(x_2) + b \ .$

The slope m is found taking the difference:

$m = \frac { \mathrm {log} (F_1) - \mathrm {log} (F_2)} { \mathrm{log}(x_1) - \mathrm{log}(x_2) } = \frac {\mathrm {log} (F_1/F_2)}{\mathrm{log}(x_1/x_2)} \ ,$

where F1 is shorthand for F ( x1 ) and the same for F2. The figure at right illustrates the formula. Notice that the slope in the example of the figure is negative. The formula also provides a negative slope, as can be seen from the following property of the logarithm:

$\mathrm{log}(x_1/x_2) = -\mathrm{log}(x_2/x_1) \ .$

Finding the function from the log-log plot

The above procedure now is reversed to find the form of the function F(x) using its (assumed) known log-log plot. To find the function F, pick some fixed point (x0, F0), where F0 is shorthand for F( x 0), somewhere on the straight line in the above graph, and further some other arbitrary point (x, F) on the same graph. Then from the slope formula above:

$m = \frac {\mathrm {log} (F/F_0)}{\mathrm{log}(x/x_0)}$

$\mathrm{log}(F / F_0) = m \,\, \mathrm{log}(x / x_0) = \mathrm{log}[(x / x_0)^m ]\ .$

Notice that 10log10( F ) = F . Therefore, the logs can be inverted to find:

$\frac{F}{F_0} = (\frac{x}{x_0})^m$

or

$F = \frac{F_0}{(x_0)^m} \,\, x^m$,

which means that

$F = \mathrm{const}\,\, x^m$

In other words, F is proportional to x to the power of the slope of the straight line of its log-log graph. Of course, the inverse is true too: any function of the form

$F = \mathrm{const} \,\, x^m$

will have a straight line as its log-log graph representation, where the slope of the line is ‘’m’’.

Semi logarithmic plots

If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi logarithmic plot. The equation for a line with an ordinate axis logarithmically scaled would be:

log(F(x)) = mx + b
F(x) = 10(mx + b) = (10mx)(10b)

The equation of a line on a plot where the abscissa axis is scaled logarithmically would be

F(x) = m log10(x) + b.

Estimating values in a diagram with logarithmic scale

One method for accurate determination of values on a logarithmic axis is as follows:

1. Measure the distance from the point on the scale to the closest decade line with lower value with a ruler.
2. Divide this distance by the length of a decade (the length between two decade lines).
3. 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:

1. Measure the distance between consecutive decades with a ruler. You can use any units provided that you are consistent.
2. Take the log (value of interest/nearest lower value decade) multiplied by the number determined in step one.
3. 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).

References

1. ^ Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space. ScienceDaily (2008-05-30). 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1416 - The Council of Constance, called by the Emperor Sigismund a supporter of Antipope John XXIII burns Jerome of Prague following Retrieved on 2008-05-31. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1279 BC - Rameses II (The Great (19th dynasty becomes pharaoh of Ancient Egypt.
which references: Stanislas, Dehaene; Véronique Izard, Elizabeth Spelke, and Pierre Pica. (2008). "Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures". Science 320 (5880): 1217. doi:10.1126/science.1156540. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.