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logarithm

Summary

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logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8. In the same fashion, since 102 = 100, then 2 = log10 100. Logarithms of the latter sort (that is, logarithms with base 10) are called common, or Briggsian, logarithms and are written simply log n.

Invented in the 17th century to speed up calculations, logarithms vastly reduced the time required for multiplying numbers with many digits. They were basic in numerical work for more than 300 years, until the perfection of mechanical calculating machines in the late 19th century and computers in the 20th century rendered them obsolete for large-scale computations. The natural logarithm (with base e ≅ 2.71828 and written ln n), however, continues to be one of the most useful functions in mathematics, with applications to mathematical models throughout the physical and biological sciences.

Properties of logarithms

Logarithms were quickly adopted by scientists because of various useful properties that simplified long, tedious calculations. In particular, scientists could find the product of two numbers m and n by looking up each number’s logarithm in a special table, adding the logarithms together, and then consulting the table again to find the number with that calculated logarithm (known as its antilogarithm). Expressed in terms of common logarithms, this relationship is given by log mn = log m + log n. For example, 100 × 1,000 can be calculated by looking up the logarithms of 100 (2) and 1,000 (3), adding the logarithms together (5), and then finding its antilogarithm (100,000) in the table. Similarly, division problems are converted into subtraction problems with logarithms: log m/n = log m − log n. This is not all; the calculation of powers and roots can be simplified with the use of logarithms. Logarithms can also be converted between any positive bases (except that 1 cannot be used as the base since all of its powers are equal to 1).

Only logarithms for numbers between 0 and 10 were typically included in logarithm tables. To obtain the logarithm of some number outside of this range, the number was first written in scientific notation as the product of its significant digits and its exponential power—for example, 358 would be written as 3.58 × 102, and 0.0046 would be written as 4.6 × 10−3. Then the logarithm of the significant digits—a decimal fraction between 0 and 1, known as the mantissa—would be found in a table. For example, to find the logarithm of 358, one would look up log 3.58 ≅ 0.55388. Therefore, log 358 = log 3.58 + log 100 = 0.55388 + 2 = 2.55388. In the example of a number with a negative exponent, such as 0.0046, one would look up log 4.6 ≅ 0.66276. Therefore, log 0.0046 = log 4.6 + log 0.001 = 0.66276 − 3 = −2.33724.

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History of logarithms

The invention of logarithms was foreshadowed by the comparison of arithmetic and geometric sequences. In a geometric sequence each term forms a constant ratio with its successor; for example, …1/1,000, 1/100, 1/10, 1, 10, 100, 1,000… has a common ratio of 10. In an arithmetic sequence each successive term differs by a constant, known as the common difference; for example, …−3, −2, −1, 0, 1, 2, 3… has a common difference of 1. Note that a geometric sequence can be written in terms of its common ratio; for the example geometric sequence given above: …10−3, 10−2, 10−1, 100, 101, 102, 103…. Multiplying two numbers in the geometric sequence, say 1/10 and 100, is equal to adding the corresponding exponents of the common ratio, −1 and 2, to obtain 101 = 10. Thus, multiplication is transformed into addition. The original comparison between the two series, however, was not based on any explicit use of the exponential notation; this was a later development. In 1620 the first table based on the concept of relating geometric and arithmetic sequences was published in Prague by the Swiss mathematician Joost Bürgi


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