Logarithm of Root, Quotient, and Reciprocal.

that is, the logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

10. Corollary. If we substitute

that is, the logarithm of any root of a number is equal to the logarithm of the number divided by the exponent of the root.

11. Corollary. The equation

gives

log. m m' = log. m + log. m',

log. m' = log. m m' - log. m;

that is, the logarithm of one factor of a product is equal to the logarithm of the product diminished by the logarithm of the other factor; or, in other words,

The logarithm of the quotient is equal to the logarithm of the dividend, diminished by the logarithm of the divisor.

12. Corollary. We have, by arts. 11 and 7,

that is, the logarithm of the reciprocal of a number is

the negative of the logarithm of the number.

Logarithms in different Systems.

13. Corollary. Since zero is the reciprocal of infinity, we have

log. 0 = log.∞

that is, the logarithm of zero is negative infinity.

14. Corollary. Since we have

b1 = b,

the logarithm of the base of a system is unity.

15. Theorem. If the logarithms of all numbers are calculated in a given system, they can be obtained for any other system by dividing the given logarithms by the logarithm of the base of the required system taken in the given system.

Demonstration. Let b be the base of the given system, and b' that of the required system; and denote by log. the logarithms in the given system, and by log. the logarithms in the required system. Taking then any number m, we have, by art. 5,

If we take the logarithms of each member of this equation

in the given system, we have, by arts. 9 and 14,

log.' m X log. b' = log. m × log. b = log. m,

Logarithms of a Power of 10.

SECTION III.

COMMON LOGARITHMS AND THEIR USES.

16. The base of the system of logarithms in common use is 10.

17. Corollary. Hence, in common logarithms, we have, by arts. 14 and 7,

that is, the logarithm of a number, which is composed of a figure 1 and cyphers, is equal to the number of places by which the figure 1 is removed from the place of units; the logarithm being positive when the figure 1 is to the left of the units' place, and negative when it is to the right of the units' place.

18. Corollary. If, therefore, a number ís between 1 and

if between 10 and

10, its log. is between 0 and 1,

if between 100 and

100, its log. is between 1 and 2, 1000, its log. is between 2 and 3,

and so on.

But if between 0,1 and 1, its log. is between

- 1 and 0,

if between 0,01 & 0,1, its log. is between - 2 and -1, and so on.

To find the Logarithm of a given Number.

Hence, if the greatest integer contained in a logarithm is called its characteristic, the characteristic of the logarithm of a number is equal to the number of places by which its first significant figure on the left is removed from the units' place, the characteristic being positive when this figure is to the left of the units' place, negative when it is to the right of the units' place, and zero when it is in the units' place.

19. Logarithms have been found of such great practical use, that much labor has been devoted to the calculation and correction of logarithmic tables. In the common tables they are given to 5, 6, or 7 places of decimals. In almost all cases, however, 5 places of decimals are sufficiently accurate; and it is, therefore, advisable to save unnecessary labor, and avoid an increased liability to error, by omitting the places which may be given beyond the first five.

20. Problem. To find the logarithm of a given number from the tables.

Solution. First. Find the characteristic by the rule of art. 18.

The characteristic is the most important part of the logarithm, and yet the unskilful are very apt to err in regard to it, not appearing to consider that an error of a single unit in its value will give a result 10 times as great or as small as it should be.

If the characteristic thus found is negative, the negative sign is usually placed above it, that this sign may not be referred to the decimal part of the loga

Finding a Logarithm.

rithm, which is always positive. But calculators are in the habit of avoiding the perplexity of a negative characteristic by subtracting its absolute value from 10, and writing the difference in its stead; and, in the use of a logarithm so written, it must not be forgotten that it exceeds the true value by 10.

Secondly. In finding the decimal part of the logarithm, the decimal point of the given number is to be wholly disregarded, and any cyphers which may precede its first significant figure on the left, or follow its last significant figure on the right, are to be omitted.

When the number thus simplified is contained within the limits of the tables, which we shall regard as extending to numbers consisting of four places, the decimal part of its logarithm is found in a horizontal line with its three first figures, and in the column below its fourth figure; the second, third, and fourth figures, when wanting, being supposed to be cyphers.

When the number consists of more than four places, and is, therefore, beyond the limits of the tables, point off its first four places on the left and consider them as integers, regarding the other places as decimals.

Care must be taken not to confound the decimal point thus introduced with the actual decimal point of the number, of which it is altogether independent.

Find, in the tables, the decimal logarithm corresponding to the integral part of the number thus pointed off; and also the difference between this logarithm and the one next above it, that is, the logarithm of the number which exceeds this integral part by