is called the logarithm of m; and since, by the preceding section, this root can be found for any value. which m may have, it follows that every number has a logarithm. The logarithm of a number is usually denoted by log. before it, or simply by the letter l

7. But the value of the logarithm varies with the value of b, and therefore the value of b, which is called the base of the system of logarithms, is of great importance; and the logarithm of a number may be defined as the exponent of the power to which the base of the system must be raised in order to produce this number.

Logarithm of Product and of Power.

8. Corollary. When the base is less than unity, it follows, from art. 3, that the logarithms of all numbers greater than unity are negative, while those of all numbers less than unity are positive.

But when, as is almost always the case, the base is greater than unity, the logarithms of all numbers greater than unity are positive, while those of all numbers less than unity are negative.

it follows, that the logarithm of unity is zero in all systems.

10. Theorem.

The sums of the logarithms of several numbers is the logarithm of their continued product.

Proof. Let the numbers be m, m', m'', &c., and let b be the base of the system; we have then

the product of which is, by art. 28,

blog. m + log. m2 + log. m2 + &c. = m m' m' &c.

Hence, by art. 7,

log. m m' m' &c. = log. m + log. m' + log. m" +&c.

11. Corollary. If the number of the factors, m, m', &c. is n, and if they are all equal to each other, we have

or

log. mmm &c. = log. m + log. m + log. m + &c.

log. m2 = n log. m ;

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.

12. 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.

13. 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.

that is, the logarithm of the reciprocal of a number

is the negative of the logarithm of the number.

Logarithms in different Systems.

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

ա

log. 0

-

log.∞ =

; ܣ

that is, the logarithm of zero is negative infinity.

16. Corollary. Since we have b1 = b,

the logarithm of the base of a system is unity.

17. 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. 7,

and

blog. m =m,

If we take the logarithms of each member of this equation in the given system, we have, by arts. 11 and 16,

Logarithms of a Power of 10.

SECTION III.

COMMON LOGARITHMS AND THEIR USES.

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

19. Corollary. Hence in common logarithms, we have, by arts. 16 and 9,

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.

20. Corollary. If, therefore, a number is

between 1 and

10, its log. is between 0 and 1, if between 10 and 100, its log. is between 1 and 2, if between 100 and 1000, its log. is between 2 and 3, and so on.