Logarithm in the context of Analytical solution

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.

In electronics, gain is a measure of the ability of a two-port circuit (often an amplifier) to increase the power or amplitude of a signal from the input to the output port by adding energy converted from some power supply to the signal. It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units ("dB gain"). A gain greater than one (greater than zero dB), that is, amplification, is the defining property of an active device or circuit, while a passive circuit will have a gain of less than one.

The term gain alone is ambiguous, and can refer to the ratio of output to input voltage (voltage gain), current (current gain) or electric power (power gain). In the field of audio and general purpose amplifiers, especially operational amplifiers, the term usually refers to voltage gain, but in radio frequency amplifiers it usually refers to power gain. Furthermore, the term gain is also applied in systems such as sensors where the input and output have different units; in such cases the gain units must be specified, as in "5 microvolts per photon" for the responsivity of a photosensor. The "gain" of a bipolar transistor normally refers to forward current transfer ratio, either h_FE ("beta", the static ratio of I_c divided by I_b at some operating point), or sometimes h_fe (the small-signal current gain, the slope of the graph of I_c against I_b at a point).

The Weber–Fechner laws are two related scientific laws in the field of psychophysics, known as Weber's law and Fechner's law. Both relate to human perception, more specifically the relation between the actual change in a physical stimulus and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell.

Ernst Heinrich Weber states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is proportional to the pre-existent stimulus," while Gustav Fechner's law is an inference from Weber's law (with additional assumptions) which states that the intensity of our sensation increases as the logarithm of an increase in energy rather than as rapidly as the increase.

The hartley (symbol Hart), also called a ban, or a dit (short for "decimal digit"), is a logarithmic unit that measures information or entropy, based on base 10 logarithms and powers of 10. One hartley is the information content of an event if the probability of that event occurring is 1⁄10. It is therefore equal to the information contained in one decimal digit (or dit), assuming a priori equiprobability of each possible value. It is named after Ralph Hartley.

If base 2 logarithms and powers of 2 are used instead, then the unit of information is the shannon or bit, which is the information content of an event if the probability of that event occurring is 1⁄2. Natural logarithms and powers of e define the nat.

In mathematics, an expression or formula (including equations and inequalities) is in closed form if it is formed with constants, variables, and a set of functions considered as basic and connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions.

A slide rule is a hand-operated mechanical calculator consisting of slidable rulers for conducting mathematical operations such as multiplication, division, exponents, roots, logarithms, and trigonometry. It is one of the simplest analog computers.

Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule is closely related to nomograms used for application-specific computations. Though similar in name and appearance to a standard ruler, the slide rule is not meant to be used for measuring length or drawing straight lines. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results.

In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation ⁠${\displaystyle \log(xy)=\log(x)+\log(y)}$⁠.

If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have highly irregular solutions. For example, the gamma function is a function that satisfies the functional equation ${\displaystyle f(x+1)=xf(x)}$ and the initial value ${\displaystyle f(1)=1.}$ There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).

In information theory, the error exponent of a channel code or source code over the block length of the code is the rate at which the error probability decays exponentially with the block length of the code. Formally, it is defined as the limiting ratio of the negative logarithm of the error probability to the block length of the code for large block lengths. For example, if the probability of error ${\displaystyle P_{\mathrm {error} }}$ of a decoder drops as ${\displaystyle e^{-n\alpha }}$, where ${\displaystyle n}$ is the block length, the error exponent is ${\displaystyle \alpha }$. In this example, ${\displaystyle {\frac {-\ln P_{\mathrm {error} }}{n}}}$ approaches ${\displaystyle \alpha }$ for large ${\displaystyle n}$. Many of the information-theoretic theorems are of asymptotic nature, for example, the channel coding theorem states that for any rate less than the channel capacity, the probability of the error of the channel code can be made to go to zero as the block length goes to infinity. In practical situations, there are limitations to the delay of the communication and the block length must be finite. Therefore, it is important to study how the probability of error drops as the block length go to infinity.

In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually ${\displaystyle t}$, in the time domain) to a function of a complex variable ${\displaystyle s}$ (in the complex-valued frequency domain, also known as s-domain or s-plane). The functions are often denoted by ${\displaystyle x(t)}$ for the time-domain representation and ${\displaystyle X(s)}$ for the frequency-domain.

The transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations, and by simplifying convolution into multiplication.