The logarithm of a number is the power to which a given number has to be raised to give a wanted result.
Example:
Given the number b = 5 and the wanted result 125. To which power has 5 to be raised to give the result 125? The answer is 3, since 53 = 125. The number b is called the base of the logarithm. The notation is log5125 = 3, or in words: The logarithm of 125 to the base 5 is 3.
The theory of logarithms can be extended to results that are fractions or irrational numbers. Indian texts showed methods to find logarithms such as log28 (which is 2 times the square root of 2). John Napier and the Swiss mathematician Joost Bürgi independently of each other developed the full theory of logarithms to any base. The logarithm to the base 10 is called the common logarithm and written log (without the base). The logarithm to the base e = 2.71828 is written ln and called the natural logarithm.
The principle use of logarithms is the conversion of complicated calculations into easier ones. Multiplication of two numbers, for example, can be achieved through the addition of their logarithms:
Given a = 12.5 and b = 5.75, what is a . b? |
The advantages of logarithms become obvious when numbers with many more digits than used in this example have to be multiplied, as is required for accurate astronomical calculations. For less accurate calculations they form the basis of the slide rule.
John Napier (b. 1550, Merchiston Castle near Edinburgh, Scotland, d. 4 April 1617, Merchiston Castle) was a Protestant theological writer and church politician who spent his leisure time on mathematical problems. He was particularly interested in methods to facilitate astronomical computations. From about 1594 he discovered the principle of logarithms and published a description of logarithms and their use in 1614 as Mirifici Logarithmorum Canonis Descriptio ("Description of the Marvelous Canon of Logarithms"). He also promised to publish an explanation how a table of logarithms can be constructed, but the corresponding publication Mirifici Logarithmorum Canonis Constructio ("Construction of the Marvelous Canon of Logarithms") appeared only four years after his death.
Joobst Bürgi (b. 28 February 1552 Lichtensteig, Switzerland, d. 31 January 1632, Kassel, Hesse-Kassel, Germany) was a mathematician who worked as the court clock maker in the royal observatory in Kassel. His fame as an inventor of astronomical instruments brought him to Prague to take up the position of imperial clockmaker to the Holy Roman Emperor Rudolf II in the new science centre. He developed his idea of logarithms from about 1588 (several years before Napier) but did not publish them until 1620. Priority of thought is therefore usually given to Napier.