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Logarithms

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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 log_a x, and, if a^y = x, then y = log_a 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, log_a a = 1 because a¹ = 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. log_a(xy) = log_a x + log_ay, so log₁₀(15) = log₁₀ 3 + log₁₀ 5

2. log_a = log_a x − log_a y, so log (⅔) = log 2 − log 3

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

4. log_a = −log_a x, so log (¼) = (−1) log 4 because ¼ = (4)^-1

5. log_a 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 (1550–1617), 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 (1561–1630) 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.