The logarithm of a number or variable x to base b, logb x, is the exponent of b needed to give x. The bases most commonly used in mathematics are e and 10. A logarithm to base e, written as log x or ln x, is known as a natural logarithm. (For example, loge10 is approximately 2.30258.) A logarithm to base 10 is written as log10 x and is known as a common logarithm.
Since a0 = 1 for any a; log 1 = 0 for all bases. In order to multiply two numbers together, one uses the fact that ax.ay = ax + y, and hence log (x.y) = log x + log y. We therefore look up the values of log x and log y in logarithmic tables, add these values, and then use the tables again to find the number whose logarithm is equal to the result of the addition. Similarly, since ax/ay = ax = y = log(x/y); and since (ax)y = axy, log xy = y.log x. Logx x = 1 since x1 = x.
To calculate in decimal numbers, logarithm tables need be compiled only for the numbers between 0 and 9.999 (in four-figure tables; in five-figure tables include 9.9999, and so on to as many figures as required). Larger numbers are expressed by adding a whole number (integer) called the characteristic, which represents in base-10 logs the corresponding power. The four-figure logarithm to the base 10 (written log10) of 2, for example, is 0.3010. The log of 200 is 2.3010 and of 2,000 is 3.3010 (200 is 102 × 100.3010 = 102.3010, and 2,000 is 103 × 100.3010 = 103.3010).
The antilogarithm of a number x is the number whose logarithm is x; that is, if log y = x, then y is the antilogarithm of x.
A logarithmic curve is the plotting of a function of the form f(x) = log x (i.e., a logarithmic function).
Logarithms, as an aid to calculation, were introduced by John Napier in 1614 and developed by Henry Briggs
Related entry slide rule
Related category FUNCTIONS
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