Log tables can be used to multiply or divide numbers. In this example 1.113 is found by adding the log under 1.11 so that for 0.003, to give 0.0464. Similarly the log of 1.456 is 0.1632. The two logs are then added to give 0.2096 and the required product is the number that has this logarithm – the number 1.620 in the table. In practice, results are found by consulting tables of antilogarithms.
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).