natural logarithm
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(mathematics) For a number x, the power to which a given base number must be raised in order to obtain x. Written \log_b x. For example, \log_{10} 1000 = 3 because 10^3 = 1000 and \log_2 16 = 4 because 2^4 = 16.
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for a number x, the power to which a given base number must be raised in order to obtain x
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1 TU was defined such that the number of TUs was ten times the base-10 logarithm of the ratio of measured power to a reference power.
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Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm.
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In other words, the amount of memory needed to store N grows logarithmically with N. Entropy is broadly a measure of the disorder of some system.
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1614: John Napier publishes Mirifici Logarithmorum Canonis Descriptio, the first table of logarithms.
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Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference.
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Landau kept a list of names of physicists which he ranked on a logarithmic scale of productivity ranging from 0 to 5.
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Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit.
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The Richter magnitude of an earthquake is determined from the logarithm of the amplitude of waves recorded by seismographs (adjustments are included to compensate for the variation in the distance between the various seismographs and the epicenter of the earthquake).
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A function f(x) is said to grow logarithmically if f(x) is (exactly or approximately) proportional to the logarithm of x.
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The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used.
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Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems.
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To quote Planck, "the logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases".
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The so-called Landauer limit implied by the laws of physics sets a lower limit on the energy required to perform a computation of kT · ln 2 per bit erased in a computation, where T is the temperature of the computing device in kelvins, k is the Boltzmann constant, and the natural logarithm of 2 is about 0.693.
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A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
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Yet another example is pH in chemistry; pH is the negative of the common logarithm of the activity of hydronium ions (the form hydrogen ions H+ take in water).
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In astrophysics, the surface gravity may be expressed as log g, which is obtained by first expressing the gravity in cgs units, where the unit of acceleration is centimeters per second squared, and then taking the base-10 logarithm.
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Finally, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same kind of quantity.
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Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations.
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Increasing education shifts this to a linear estimate (positioning 1000 10x as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.
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It is a base-10 logarithmic scale obtained by calculating the logarithm of the combined horizontal amplitude of the largest displacement from zero on a Wood–Anderson torsion seismometer output.
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The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.
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The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms.
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He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms.
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For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential.
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Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.
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