Higher Algebra

In this and the three preceding articles the mantissæ have been supposed positive. In order to secure the advantages of Briggs' system, we arrange our work so as always to keep the mantissa positive, so that when the mantissa of any logarithm has been taken from the Tables the characteristic is prefixed with its appropriate sign according to the rules already given.

214. In the case of a negative logarithm the minus sign is written over the characteristic, and not before it, to indicate that the characteristic alone is negative, and not the whole expression. Thus 4.30103, the logarithm of 0002, is equivalent to −4 + ·30103, and must be distinguished from - 4.30103, an expression in which both the integer and the decimal are negative. In working with negative logarithms an arithmetical artifice will sometimes be necessary in order to make the mantissa positive. For instance, a result such as - 3.69897, in which the whole expression is negative, may be transformed by subtracting 1 from the characteristic and adding 1 to the mantissa. Thus

Example 1. Required the logarithm of 0002432.

In the Tables we find that 3859636 is the mantissa of log 2432 (the decimal point as well as the characteristic being omitted); and, by Art. 211, the characteristic of the logarithm of the given number is -4;

the mantissa of log 00000165 being the same as that of log 165, and the characteristic being prefixed by the rule.

and 8434968 is the mantissa of log 697424; hence x is a number consisting of these same digits but with one cipher after the decimal point. [Art. 211.]

215. The method of calculating logarithms will be explained in the next chapter, and it will there be seen that they are first found to another base, and then transformed into common logarithms to base 10.

It will therefore be necessary to investigate a method for transforming a system of logarithms having a given base to a new system with a different base.

216. Suppose that the logarithms of all numbers to base a are known and tabulated, it is required to find the logarithms to base b.

Let N be any number whose logarithm to base b is required.

Now since N and b are given, log N and log b are known from the Tables, and thus log, may be found.

Hence it appears that to transform logarithms from base a to base b we have only to multiply them all by this is a log.b; constant quantity and is given by the Tables; it is known as the modulus.

thus

217. In equation (1) of the preceding article put a for N;

This result may also be proved directly as follows:

Let

x=log.b, so that a* = b;

then by taking logarithms to base b, we have

218. The following examples will illustrate the utility of logarithms in facilitating arithmetical calculation; but for information as to the use of Logarithmic Tables the reader is referred to works on Trigonometry.