Similarly, division problems are converted into subtraction problems with logarithms: log m/ n = 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. Expressed in terms of common logarithms, this relationship is given by log m n = log m + log n. 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). Logarithms were quickly adopted by scientists because of various useful properties that simplified long, tedious calculations.
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. 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. Invented in the 17th century to speed up calculations, logarithms vastly reduced the time required for multiplying numbers with many digits. Logarithms of the latter sort (that is, logarithms with base 10) are called common, or Briggsian, logarithms and are written simply log n. For example, 2 3 = 8 therefore, 3 is the logarithm of 8 to base 2, or 3 = log 2 8. Expressed mathematically, x is the logarithm of n to the base b if b x = n, in which case one writes x = log b n. Logarithm, the exponent or power to which a base must be raised to yield a given number.
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