A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences between the magnitudes of the numbers involved.
Unlike a linear scale where each unit of distance corresponds to the same increment, on a logarithmic scale each unit of length is a multiple of some base value raised to a power, and corresponds to the multiplication of the previous value in the scale by the base value.
A logarithmic scale is nonlinear, and as such numbers with equal distance between them such as 1, 2, 3, 4, 5 are not equally spaced. Equally spaced values on a logarithmic scale have exponents that increment uniformly. Examples of equally spaced values are 10, 100, 1000, 10000, and 100000 (i.e., 10^1, 10^2, 10^3, 10^4, 10^5) and 2, 4, 8, 16, and 32 (i.e., 2^1, 2^2, 2^3, 2^4, 2^5).
Exponential growth curves are often depicted on a logarithmic scale to prevent them from expanding too rapidly and becoming too large to fit within a small graph.
The markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.
The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
Some of our senses operate in a logarithmic fashion (Weber–Fechner law), 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. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers in some cultures. [1]
The top left graph is linear in the X and Y axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the bottom left graph, and the Y axis ranges from 0.1 to 1,000.
The top right graph uses a log-10 scale for just the X axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y axis.
Presentation of data on a logarithmic scale can be helpful when the data:
A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool.
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot.
If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.
A modified log transform can be defined for negative input (y<0) to avoid the singularity for zero input (y=0), and so produce symmetric log plots: [2] [3]
for a constant C=1/ln(10).
A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.
Examples of logarithmic units include units of information and information entropy (nat, shannon, ban) and of signal level (decibel, bel, neper). Frequency levels or logarithmic frequency quantities have various units are used in electronics (decade, octave) and for music pitch intervals (octave, semitone, cent, etc.). Other logarithmic scale units include the Richter magnitude scale point.
In addition, several industrial measures are logarithmic, such as standard values for resistors, the American wire gauge, the Birmingham gauge used for wire and needles, and so on.
Unit | Base of logarithm | Underlying quantity | Interpretation |
---|---|---|---|
bit | 2 | number of possible messages | quantity of information |
byte | 28 = 256 | number of possible messages | quantity of information |
decibel | 10(1/10) ≈ 1.259 | any power quantity (sound power, for example) | sound power level (for example) |
decibel | 10(1/20) ≈ 1.122 | any root-power quantity (sound pressure, for example) | sound pressure level (for example) |
semitone | 2(1/12) ≈ 1.059 | frequency of sound | pitch interval |
The two definitions of a decibel are equivalent, because a ratio of power quantities is equal to the square of the corresponding ratio of root-power quantities.[ citation needed ]
The decibel is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a power ratio of 101/10 or root-power ratio of 101⁄20.
In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is 10, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of some powers of 2, since computers store data in a binary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value.
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.
The neper is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.
In electronics, gain is a measure of the ability of a two-port circuit 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. A gain greater than one, that is, amplification, is the defining property of an active component or circuit, while a passive circuit will have a gain of less than one.
In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.
The phon is a logarithmic unit of loudness level for tones and complex sounds. Loudness is measured in sones, a linear unit. Human sensitivity to sound is variable across different frequencies; therefore, although two different tones may present an identical sound pressure to a human ear, they may be psychoacoustically perceived as differing in loudness. The purpose of the phon is to provide a logarithmic measurement for perceived sound magnitude, while the primary loudness standard methods result in a linear representation. A sound with a loudness of 1 sone is judged equally loud as a 1 kHz tone with a sound pressure level of 40 decibels above 20 micropascal. The phon is psychophysically matched to a reference frequency of 1 kHz. In other words, the phon matches the sound pressure level (SPL) in decibels of a similarly perceived 1 kHz pure tone. For instance, if a sound is perceived to be equal in intensity to a 1 kHz tone with an SPL of 50 dB, then it has a loudness of 50 phons, regardless of its physical properties. The phon was proposed in DIN 45631 and ISO 532 B by Stanley Smith Stevens.
In mathematics, the binary logarithm is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x,
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.
This is a list of logarithm topics, by Wikipedia page. See also the list of exponential topics.
In industrial design, preferred numbers are standard guidelines for choosing exact product dimensions within a given set of constraints. Product developers must choose numerous lengths, distances, diameters, volumes, and other characteristic quantities. While all of these choices are constrained by considerations of functionality, usability, compatibility, safety or cost, there usually remains considerable leeway in the exact choice for many dimensions.
One decade is a unit for measuring ratios on a logarithmic scale, with one decade corresponding to a ratio of 10 between two numbers.
In science and engineering, a semi-log plot/graph or semi-logarithmicplot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range of values, or to zoom in and visualize that - what seems to be a straight line in the beginning - is in fact the slow start of a logarithmic curve that is about to spike and changes are much bigger than thought initially.
In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point zi is replaced with the transformed value yi = f(zi), 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.
The International System of Quantities (ISQ) is a standard system of quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities.
In electronics, an octave is a logarithmic unit for ratios between frequencies, with one octave corresponding to a doubling of frequency. For example, the frequency one octave above 40 Hz is 80 Hz. The term is derived from the Western musical scale where an octave is a doubling in frequency. Specification in terms of octaves is therefore common in audio electronics.
In statistics, a misleading graph, also known as a distorted graph, is a graph that misrepresents data, constituting a misuse of statistics and with the result that an incorrect conclusion may be derived from it.
In science and engineering, a power level and a field level are logarithmic magnitudes of certain quantities referenced to a standard reference value of the same type.
In mathematics, the set of positive real numbers, is the subset of those real numbers that are greater than zero. The non-negative real numbers, also include zero. Although the symbols and are ambiguously used for either of these, the notation or for and or for has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.