Logarithm Totally Explained

Everything about Logarithm totally explained

In mathematics, the logarithm of a given number to a given base is the power or exponent to which the base must be raised in order to produce the given number.
For example, the logarithm of 1000 to the common base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.
The logarithm of x to the base b is written log_b(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y, ť

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However, it can also be computed in O(log n) steps by trying to shift by powers of 2 and checking that the result stays nonzero: for example, first >>16, then >>8, ... (Each step reveals one bit of the result)

Generalizations

The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it's a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus of z, arg(z) is the argument, and i is the imaginary unit; see complex logarithm for details. The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bⁿ = x, where b and x are elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it's believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in public key cryptography. The logarithm of a matrix is the inverse of the matrix exponential. It is possible to take the logarithm of a quaternions and octonions. A double logarithm,

ln(ln(x))

, is the inverse function of the double exponential function. A super-logarithm or hyper-4-logarithm is the inverse function of tetration. The super-logarithm of x grows even more slowly than the double logarithm for large x.
For each positive b not equal to 1, the function log_b (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.

History

   The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston, in Scotland. (Joost Bürgi independently discovered logarithms; however, he didn't publish his discovery until four years after Napier.) Early resistance to the use of logarithms was muted by Kepler's enthusiastic support and his publication of a clear and impeccable explanation of how they worked.
   Their use contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, they were used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides the utility of the logarithm concept in computation, the natural logarithm presented a solution to the problem of quadrature of a hyperbolic sector at the hand of Gregoire de Saint-Vincent in 1647.
   At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers they represent, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.
   Napier didn't use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it's useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10⁻⁷ = 0.999999 (Bürgi chose r = 1 + 10⁻⁴ = 1.0001). Napier's original logarithms didn't have log 1 = 0 but rather log 10⁷ = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 10⁷(1 − 10⁻⁷)^L. Since (1 − 10⁻⁷)^10⁷ is approximately 1/e, this makes L/10⁷ approximately equal to log_1/e N/10⁷. An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega. François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.
   Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.
   Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1700s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

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