What are the connectionsbetween
logarithms and algebra?
Logarithms are the numbers of the power
to which a base must be raised in order to
get a given positive number. For example,
the logarithm of 100 to the base 10 is 2,
or log 10100 2. This is because 10^2
- Common logarithmsare positive
numbers that use the number 10 as the
base; they are written as log x. Those
using the number symbolized by eas the
base are called natural logarithms(also
phrased as logarithms with a base e); the
natural logarithm of a number xis writ-
ten as ln x.
What’s the connection? Because loga-
rithms are really exponents, they satisfy
all the usual rules of exponents. Conse-
quently, tedious and long algebraic calcu-
lations such as those involving multipli-
cation and division can be replaced by the
simpler processes of adding or subtracting the corresponding logarithms. In general,
logarithmic tables are usually used for this purpose—although calculators, comput-
ers, and the Internet often replace the need for such tables.
What was the progressionof logarithm development?
The invention of logarithms was a long process, starting with Scottish mathematician
John Napier (1550–1617; also known as Laird of Merchiston), who first came up with
the idea of logarithms in 1594. But the actual invention and announcement of loga-
rithms would take another 20 years: In 1614, Napier would publish Mirifici logarith-
morum canonis descripto(Description of the Wonderful Canon of Logarithms), which
offered tables and rules for their use.
Not long afterward, in 1617, English mathematician Henry Briggs (1561–1630)
published Logarithmorum chilias prima(Logarithms of Numbers from 1 to 1,000),
introducing the concept of common logarithms—or logarithms based on the powers
of ten. And finally, independently from Briggs and Napier, came Swiss mathematician
Joost Bürgi (1552–1632), who in 1620 presented Arithmetische und geometrische
Progress-tabulen,a German work presenting the discovery of logarithms.
These discoveries differed in several ways: Napier’s approach was algebraic; Bürgi’s
was geometric. There were differences from the common and natural logarithms we 145
ALGEBRA
Even the simple act of cranking up your sound sys-
tem involves math. The decibel scale used by your
loudspeakers and amplifiers employs the concept of
logarithms. The Image Bank/Getty Images.