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