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=== Log ===
In [https://en.wikipedia.org/wiki/Mathematics mathematics], the '''logarithm''' is the [https://en.wikipedia.org/wiki/Inverse_function inverse function] to [https://en.wikipedia.org/wiki/Exponentiation exponentiation]. That means that the logarithm of a number x to the '''base''' b is the [https://en.wikipedia.org/wiki/Exponent exponent] to which b must be raised to produce x. For example, since 1000 = 10<sup>3</sup>, the ''logarithm base'' 10<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0> of 1000 is 3, or log<sub>10</sub> (1000) = 3. The logarithm of x to ''base'' b is denoted as log<sub>''b''</sub> (''x''), or without parentheses, log<sub>''b''</sub> ''x''. When the base is clear from the context or is irrelevant, such as in [https://en.wikipedia.org/wiki/Big_O_notation big O notation], it is sometimes written log ''x''. The logarithm base 10 is called the ''decimal'' or [https://en.wikipedia.org/wiki/Common_logarithm ''common'' logarithm] and is commonly used in science and engineering. The [https://en.wikipedia.org/wiki/Natural_logarithm ''natural'' logarithm] has the number [https://en.wikipedia.org/wiki/E_(mathematical_constant) ''e'' ≈ 2.718] as its base; its use is widespread in mathematics and [https://en.wikipedia.org/wiki/Physics physics], because of its very simple [https://en.wikipedia.org/wiki/Derivative derivative]. The [https://en.wikipedia.org/wiki/Binary_logarithm ''binary'' logarithm] uses base 2 and is frequently used in [https://en.wikipedia.org/wiki/Computer_science computer science]. Logarithms were introduced by [https://en.wikipedia.org/wiki/John_Napier John Napier] in 1614 as a means of simplifying calculations.<sup id="cite_ref-1">[https://en.wikipedia.org/wiki/Logarithm#cite_note-1 [1]]</sup> They were rapidly adopted by navigators, scientists, engineers, [https://en.wikipedia.org/wiki/Surveying surveyors], and others to perform high-accuracy computations more easily. Using [https://en.wikipedia.org/wiki/Mathematical_table#Tables_of_logarithms logarithm tables], tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a [https://en.wikipedia.org/wiki/Product_(mathematics) product] is the [https://en.wikipedia.org/wiki/Summation sum] of the logarithms of the factors: log𝑏(𝑥𝑦)=log𝑏𝑥+log𝑏𝑦,<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/72599165912508b07108f2a840898022ed126148><br/> provided that b, x and y are all positive and ''b'' ≠ 1. The [https://en.wikipedia.org/wiki/Slide_rule slide rule], also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from [https://en.wikipedia.org/wiki/Leonhard_Euler Leonhard Euler], who connected them to the [https://en.wikipedia.org/wiki/Exponential_function exponential function] in the 18th century, and who also introduced the letter e as the base of natural logarithms.<sup id="cite_ref-2">[https://en.wikipedia.org/wiki/Logarithm#cite_note-2 [2]]</sup> [https://en.wikipedia.org/wiki/Logarithmic_scale Logarithmic scales] reduce wide-ranging quantities to smaller scopes. For example, the [https://en.wikipedia.org/wiki/Decibel decibel] (dB) is a [https://en.wikipedia.org/wiki/Units_of_measurement unit] used to express [https://en.wikipedia.org/wiki/Level_(logarithmic_quantity) ratio as logarithms], mostly for signal power and amplitude (of which [https://en.wikipedia.org/wiki/Sound_pressure sound pressure] is a common example). In chemistry, [https://en.wikipedia.org/wiki/PH pH] is a logarithmic measure for the [https://en.wikipedia.org/wiki/Acid acidity] of an [https://en.wikipedia.org/wiki/Aqueous_solution aqueous solution]. Logarithms are commonplace in scientific [https://en.wikipedia.org/wiki/Formula formulae], and in measurements of the [https://en.wikipedia.org/wiki/Computational_complexity_theory complexity of algorithms] and of geometric objects called [https://en.wikipedia.org/wiki/Fractal fractals]. They help to describe [https://en.wikipedia.org/wiki/Frequency frequency] ratios of [https://en.wikipedia.org/wiki/Interval_(music) musical intervals], appear in formulas counting [https://en.wikipedia.org/wiki/Prime_number prime numbers] or [https://en.wikipedia.org/wiki/Stirling's_approximation approximating] [https://en.wikipedia.org/wiki/Factorial factorials], inform some models in [https://en.wikipedia.org/wiki/Psychophysics psychophysics], and can aid in [https://en.wikipedia.org/wiki/Forensic_accounting forensic accounting]. The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the [https://en.wikipedia.org/wiki/Complex_logarithm complex logarithm] is the multi-valued [https://en.wikipedia.org/wiki/Inverse_function inverse] of the complex exponential function. Similarly, the [https://en.wikipedia.org/wiki/Discrete_logarithm discrete logarithm] is the multi-valued inverse of the exponential function in finite groups; it has uses in [https://en.wikipedia.org/wiki/Public-key_cryptography public-key cryptography].<br/> <br/> full text link : [https://en.wikipedia.org/wiki/Logarithm https://en.wikipedia.org/wiki/Logarithm]
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