| 释义 |
LogarithmThe logarithm is defined to be the Inverse Function of taking a number to a given Power. Therefore, for any  and  ,
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 | (2) |
 . In the above plot, the solid curve is the logarithm toBase  (the Natural Logarithm), and the dotted curve is the logarithm to Base 10 (Log).
Logarithms are used in many areas of science and engineering in which quantities vary over a large range. For example, the decibelscale for the loudness of sound, the Richter scale of earthquake magnitudes, and the astronomical scale of stellar brightnesses areall logarithmic scales.
The logarithm can also be defined for Complex arguments, as shown above. If the logarithm is taken asthe forward function, the function taking the Base to a given Power is then called theAntilogarithm.
For  ,  is called the Characteristic and  is called theMantissa. Division and multiplication identities follow from these
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 | (4) |
 | (5) |
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 | (8) |
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The logarithm Base e is called the Natural Logarithm and is denoted  (Ln). The logarithm Base 10 is denoted  (Log), (although mathematics textsoften use to mean ). The logarithm Base 2 is denoted  (Lg).
An interesting property of logarithms follows from looking for a number  such that
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 | (13) |
 | (14) |
 | (15) |
 | (16) |
Numbers of the form  are Irrational if  and are Integers, one of which hasa Prime factor which the other lacks. A. Baker made a major step forward in Transcendental Number theory byproving the transcendence of sums of numbers of the form  for  and  AlgebraicNumbers. See also Antilogarithm, Cologarithm, e, Exponential Function, Harmonic Logarithm,Lg, Ln, Log, Logarithmic Number, Napierian Logarithm, Natural Logarithm,Power
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Logarithmic Function.'' §4.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 67-69, 1972.Conway, J. H. and Guy, R. K. ``Logarithms.'' The Book of Numbers. New York: Springer-Verlag, pp. 248-252, 1996. Beyer, W. H. ``Logarithms.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 159-160, 1987. Pappas, T. ``Earthquakes and Logarithms.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 20-21, 1989. Spanier, J. and Oldham, K. B. ``The Logarithmic Function  .'' Ch. 25 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 225-232, 1987. |
