logarithm (redirected from Napierian logarithm)

logarithm (lŏg`ərĭthəm) [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.

---

logarithm

In mathematics, the power to which a base must be raised to yield a given number (e.g., the logarithm to the base 3 of 9, or log₃ 9, is 2, because 3² = 9). A common logarithm is a logarithm to the base 10. Thus, the common logarithm of 100 (log 100) is 2, because 10² = 100. Logarithms to the base e, in which e = 2.71828…, called natural logarithms (ln), are especially useful in calculus. Logarithms were invented to simplify cumbersome calculations, since exponents can be added or subtracted to multiply or divide their bases. These processes have been further simplified by the incorporation of logarithmic functions into digital calculators and computers. See also John Napier.