Logarithm in the context of "American wire gauge"

In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable ${\displaystyle X}$, which may be any member ${\displaystyle x}$ within the set ${\displaystyle {\mathcal {X}}}$ and is distributed according to ${\displaystyle p\colon {\mathcal {X}}\to [0,1]}$, the entropy is$${\displaystyle \mathrm {H} (X):=-\sum _{x\in {\mathcal {X}}}p(x)\log p(x),}$$where ${\displaystyle \Sigma }$ denotes the sum over the variable's possible values. The choice of base for ${\displaystyle \log }$, the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannons"), while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.

The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication", and is also referred to as Shannon entropy. Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel, and a receiver. The "fundamental problem of communication" – as expressed by Shannon – is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. Shannon considered various ways to encode, compress, and transmit messages from a data source, and proved in his source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his noisy-channel coding theorem.

Shipbuilding is the construction of ships and other floating vessels. In modern times, it normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation that traces its roots to before recorded history.

Until recently, with the development of complex non-maritime technologies, a ship has often represented the most advanced structure that the society building it could produce. Some key industrial advances were developed to support shipbuilding, for instance the sawing of timbers by mechanical saws propelled by windmills in Dutch shipyards during the first half of the 17th century. The design process saw the early adoption of the logarithm (invented in 1615) to generate the curves used to produce the shape of a hull, especially when scaling up these curves accurately in the mould loft.

In ecology, the species discovery curve (also known as a species accumulation curve or collector's curve) is a graph recording the cumulative number of species of living things recorded in a particular environment as a function of the cumulative effort expended searching for them (usually measured in person-hours). It is related to, but not identical with, the species-area curve.

The species discovery curve will necessarily be increasing, and will normally be negatively accelerated (that is, its rate of increase will slow down). Plotting the curve gives a way of estimating the number of additional species that will be discovered with further effort. This is usually done by fitting some kind of functional form to the curve, either by eye or by using non-linear regression techniques. Commonly used functional forms include the logarithmic function and the negative exponential function. The advantage of the negative exponential function is that it tends to an asymptote which equals the number of species that would be discovered if infinite effort is expended. However, some theoretical approaches imply that the logarithmic curve may be more appropriate, implying that though species discovery will slow down with increasing effort, it will never entirely cease, so there is no asymptote, and if infinite effort was expended, an infinite number of species would be discovered. An example in which one would not expect the function to asymptote is in the study of genetic sequences where new mutations and sequencing errors may lead to infinite variants.

The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The classification of the series by the Rydberg formula was important in the development of quantum mechanics. The spectral series are important in astronomical spectroscopy for detecting the presence of hydrogen and calculating red shifts.

Soil pH is a measure of the acidity or basicity (alkalinity) of a soil. Soil pH is a key characteristic that can be used to make informative analysis both qualitative and quantitatively regarding soil characteristics. pH is defined as the negative logarithm (base 10) of the activity of hydronium ions (H
or, more precisely, H
₃O
_aq) in a solution. In soils, it is measured in a slurry of soil mixed with water (or a salt solution, such as 0.01 M CaCl
₂), and normally falls between 3 and 10, with 7 being neutral. Acid soils have a pH below 7 and alkaline soils have a pH above 7. Ultra-acidic soils (pH < 3.5) and very strongly alkaline soils (pH > 9) are rare.

Soil pH is considered a master variable in soils as it affects many chemical processes. It specifically affects plant nutrient availability by controlling the chemical forms of the different nutrients and influencing the chemical reactions they undergo. The optimum pH range for most plants is between 5.5 and 7.5; however, many plants have adapted to thrive at pH values outside this range.

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, to check intonation, or to compare the sizes of comparable intervals in different tuning systems. For humans, a single cent is too small to be perceived between successive notes.

Cents, as described by Alexander John Ellis, follow a tradition of measuring intervals by logarithms that began with Juan Caramuel y Lobkowitz in the 17th century. Ellis chose to base his measures on the hundredth part of a semitone, ${\displaystyle {\sqrt[{1200}]{2}}}$, at Robert Holford Macdowell Bosanquet's suggestion. Making extensive measurements of musical instruments from around the world, Ellis used cents to report and compare the scales employed, and further described and utilized the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals.

In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point z_i is replaced with the transformed value y_i = f(z_i), where f is a function. Transforms are usually applied so that the data appear to more closely meet the assumptions of a statistical inference procedure that is to be applied, or to improve the interpretability or appearance of graphs.

Nearly always, the function that is used to transform the data is invertible, and generally is continuous. The transformation is usually applied to a collection of comparable measurements. For example, if we are working with data on peoples' incomes in some currency unit, it would be common to transform each person's income value by the logarithm function.

Equisetum (/ˌɛkwɪˈsiːtəm/; horsetail) is the only living genus in Equisetaceae, a family of vascular plants that reproduce by spores rather than seeds.

Equisetum is a "living fossil", the only living genus of the entire subclass Equisetidae, which for over 100 million years was much more diverse and dominated the understorey of late Paleozoic forests. Some equisetids were large trees reaching to 30 m (98 ft) tall. The genus Calamites of the family Calamitaceae, for example, is abundant in coal deposits from the Carboniferous period. The pattern of spacing of nodes in horsetails, wherein those toward the apex of the shoot are increasingly close together, is said to have inspired John Napier to invent logarithms. Modern horsetails first appeared during the Jurassic period.