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Logarithms

Logarithms


The logarithm of a positive real number x to the base-a is the number y that satisfies the equation a y = x. In symbols, the logarithm of x to the base-a is loga x, and, if a y = x, then y = loga x.

Essentially, the logarithm to base-a is a function: To each positive real number x, the logarithm to base-a assigns x a number y such that a y = x. For example, 102 = 100; therefore, log10 100 = 2. The logarithm of 100 to base-10 is 2, which is an elaborate name for the power of 10 that equals 100.

Any positive real number except 1 can be used as the base. However, the two most useful integer bases are 10 and 2. Base-2 , also known as the binary system, is used in computer science because nearly all computers and calculators use base-2 for their internal calculations. Logarithms to the base-10 are called common logarithms. If the base is not specified, then base-10 is assumed, in which case the notation is simplified to log x.

Some examples of logarithms follow.

log 1 = 0 because 100 = 1

log 10 = 1 because 101 = 10

log 100 = 2 because 102 = 100

log2 8 = 3 because 23 = 8

log2 2 = 1 because 21 = 2

log5 25 = 2 because 52 = 25

log3 = 2 because 3-2 =

The logarithm of multiples of 10 follows a simple pattern: logarithm of 1,000, 10,000, and so on to base-10 are 3, 4 and so on. Also, the logarithm of a number a to base-a is always 1; that is, loga a = 1 because a 1 = a.

Logarithms have some interesting and useful properties. Let x, y, and a be positive real numbers, with a not equal to 1. The following are five useful properties of logarithms.

1. loga(xy ) = loga x + logay, so log10(15) = log10 3 + log10 5

2. loga = loga x loga y, so log () = log 2 log 3

3. loga x r = r loga x, where r is any real number, so log 35 = 5 log 3

4. loga = loga x, so log (¼) = (1) log 4 because ¼ = (4)-1

5. loga a r = r, so log10 103 = 3

Logarithms are useful in simplifying tedious calculations because of these properties.

History of Logarithms

The beginning of logarithms is usually attributed to John Napier (15501617), a Scottish amateur mathematician. Napier's interest in astronomy required him to do tedious calculations. With the use of logarithms, he developed ideas that shortened the time to do long and complex calculations. However, his approach to logarithms was different from the form used today.

Fortunately, a London professor, Henry Briggs (15611630) became interested in the logarithm tables prepared by Napier. Briggs traveled to Scotland to visit Napier and discuss his approach. They worked together to make improvements such as introducing base-10 logarithms. Later, Briggs developed a table of logarithms that remained in common use until the advent of calculators and computers. Common logarithms are occasionally also called Briggsian logarithms.

see also Powers and Exponents.

Rafiq Ladhani

Bibliography

James, Robert C., and Glenn James. Mathematics Dictionary, 5th ed. New York: Van Nostrand Reinhold, 1992.

Young, Robyn V., ed. Notable Mathematicians, from Ancient Times to the Present. Detroit: Gale Research, 1998.

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Logarithm

Logarithm

In the 1500s and early 1600s, although science, engineering, and medicine were flourishing, many people did not understand multiplication tables. Mathematicians, astronomers, navigators, and scientists were forced to spend a lot of time performing calculations, so that little time was left to work on experiments and new discoveries. Finally, around 1594 Scottish mathematician John Napier (15501617) produced a table of logarithmic, or proportionate, numbers.

How logarithms work

In the commonly known base 10 system, computations that involve very large numbers can become difficult, if not incomprehensible. Napier realized numbers could be more easily expressed in terms of powers. Thus 100 is equal to 10 multiplied by 10, written as 102. This is read as "10 squared" and means "10 to the power two."

To perform multiplication, numbers are converted into logarithms, the exponents added together, and the result converted back into base 10. Likewise, to perform division, two logarithmic exponents are subtracted, and the result converted back to base 10.

This innovative way of multiplying and dividing large numbers was a milestone event for mathematicians of the day. Napier's tables were published in 1614 and were put into use immediately, becoming an essential part of the mathematical, scientific, and navigational processes.

Logarithmic tables remained popular throughout the next several centuries and were used as the basis for many mechanical calculating devices. Relieved from much of their mental drudgery, scientists and mathematicians enjoyed new freedom in their work, allowing them to focus their attention on new scientific breakthroughs.

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logarithm

logarithm (lŏg´ərĬŧħəm) [Gr.,=relation number], number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number. For example, the logarithm of 100 to the base 10 is 2, written log10 100=2, since 102=100. Logarithms of positive numbers using the number 10 as the base are called common logarithms; those using the number e (see separate article) as the base are called natural logarithms or Napierian logarithms (for John Napier). The natural logarithm of a number x is denoted by ln x or simply log x. Since logarithms are exponents, they satisfy all the usual rules of exponents. Consequently, tedious calculations such as multiplications and divisions can be replaced by the simpler processes of adding or subtracting the corresponding logarithms. Logarithmic tables are generally used for this purpose.

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logarithm

logarithm Aid to calculation devised by John Napier in 1614, and developed by the English mathematician Henry Briggs. A number's logarithm is the power to which a base must be raised to equal the number, i.e. if bx = n, then logb n = x, where n is the number, b the base and x the logarithm. Common logarithms have base 10, and so-called natural logarithms have base e (2.71828…). Logarithms to the base 2 are used in computer science and information theory.

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logarithm

log·a·rithm / ˈlôgəˌri[voicedth]əm; ˈlägə-/ (abbr.: log) • n. a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

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logarithm

logarithm XVII. — modL. logarithmus, f. Gr. lógos ratio + arithmós number (cf. LOGOS, ARITHMETIC).

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logarithm

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