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Encyclopedia > Logarithmic scale    Various scales: lin-lin, lin-log, log-lin and log-log. Plotted graphs are: y=x (green), y=10x(red), y=log(x) (blue)

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 (doubling the input strength adds a constant to the subjective signal strength), 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. Senses are the physiological methods of perception. ... Hearing, or audition, is one of the traditional five senses, and refers to the ability to detect sound. ...

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. Logarithmic units are generic mathematical units in which we can express any quantities (physical or mathematical) that are defined as being proportional to values of a logarithm function. ...

On most logarithmic scales, small values (or ratios) of the underlying quantity correspond to small (possibly negative) values of the logarithmic measure. Well-known examples of such scales are:

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: Richter magnitude test scale (or more correctly local magnitude ML scale) assigns a single number to quantify the size of an earthquake. ... Global earthquake epicenters, 1963&#8211;1998. ... Motion involves change in position, such as this perspective of rapidly leaving Yongsan Station In physics, motion means a continuous change in the position of a body relative to a reference point, as measured by a particular observer in a particular frame of reference. ... Earth (IPA: , often referred to as the Earth, Terra, the World or Planet Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth largest. ... The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. ... For Neper as a mythological god, see Neper (mythology). ... The cent is a logarithmic unit of measure used for musical intervals. ... A minor second is the smallest of three commonly occuring musical intervals that span two diatonic scale degrees; the others being the major second and the augmented second, which are larger by one and two semitones respectively. ... A major second is one of three commonly occuring musical intervals that span two diatonic scale degrees; the others being the minor second, which is one semitone smaller, and the augmented second, which is one semitone larger. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... For other uses, see Music (disambiguation). ... In mathematics, especially as applied in statistics, the logit (pronounced with a long o and a soft g, IPA ) of a number p between 0 and 1 is This function is used in logistic regression. ... In probability theory and statistics the odds in favor of an event or a proposition are the quantity p / (1 âˆ’ p), where p is the probability of the event or proposition. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... 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. ... A logarithmic timeline, based on logarithmic scale, was developed by Heinz von Foerster, the philosopher and physicist. ... A 35mm lens set to f/11, as indicated by the white dot above the f-stop scale on the aperture ring In photography the f-number (focal ratio) expresses the diameter of the diaphragm aperture in terms of the effective focal length of the lens. ... In photography, exposure is the total amount of light allowed to fall on the film during the process of taking a photograph. ... The word probability derives from the Latin probare (to prove, or to test). ... Ice melting - classic example of entropy increasing described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Information is the result of processing, manipulating and organizing data in a way that adds to the knowledge of the person receiving it. ... A bundle of optical fiber. ...

A logarithmic scale is also a graphic 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. The slide rule (often nicknamed a slipstick) is a mechanical analog computer, consisting of at least two finely divided scales (rules), most often a fixed outer pair and a movable inner one, with a sliding window called the cursor. ... A nomogram or nomograph is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a function. ... 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 a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ...

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 semilog graph, log-log graph). In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. ... See Also: Watt In physics, a power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ... In science and engineering, a semi-log graph or semi-log plot is a way of visualizing data that is changing with an exponential relationship. ... A log-log plot of y=x (green), y=x^2 (blue), and y=x^3 (red). ...

## Estimating values in a diagram with logarithmic scale

To estimate the value of a point on a logarithmic axis:

Now this might be a tricky task to do if you want to get a value that doesn't differ too much from reality. For some uses an educated guess simply can't give you decent enough error margins (i.e. engineering purposes). The easy way to get a more correct value is the following:

1. Measure the distance from the point on your scale to the nearest decade line with lower value.
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 10^a where a is the value you found in step 2.

Example: What is the value that lies halfway between the 10 and 100 decades on a logarithmic axis? Since we are interested in the halfway point, 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 (10^0.5)*10.

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 (english or metric). You just need to be consistent with your intervals.
2. Take the log(value of interest/nearest lower value decade) multiplied by the number you 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: Let's say you want to know where 17 is located on a logarithmic axis. Use your ruler to measure the distance between 10 and 100. Let's say number of intervals is 30 on my ruler (note it can vary - you just need to be using the same scale through the rest of the process).

You take the log (17/10) * 30 = 6.9

So, on the interval 7 of the 30 intervals I counted across the decade, you will find that is where 17 belongs. Results from FactBites:

 Epic Disasters - Richter Scale (230 words) The scale is derived from astronomy's stellar magnitude scale. The number is derived by calculating the logarithm of the combined horizontal amplitude of the largest displacement from zero on a seismogram. The energy of the earthquake is proportional to the square root of the cube of the amplitude.
 Logarithmic scale - Wikipedia, the free encyclopedia (421 words) A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. Logarithmic scales are either defined for ratios of the underlying quantity, or one has to agree to measure the quantity in fixed units. A logarithmic scale is also a graphic 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.
More results at FactBites »

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