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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
log. m m m &c. — log. m+log. m + log. m +&c.
log. man 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.
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
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.
14. Corollary. We have, by arts. 13 and 9,
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
that is, the logarithm of zero is negative infinity.
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,
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,
log.'m log. b' = log m × log. b =
Logarithms of a Power of 10.
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
if between 10 and
10, its log. is between 0 and 1, 100, its log. is between 1 and 2,
if between 100 and 1000, its log. is between 2 and 3, and so on.
To find the Logarithm of a given Number.
But if between 0.1 and 1, its log. is between 1 and if between 0.01 and 0.1, its log. is between 2 and −1, and so on.
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.
21. 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.
To find the logarithm of a given number from the tables.
Solution. First. Find the characteristic by the rule of art. 20.
The characteristic is the most important part of the logarithm, and yet the unskilful are very apt to err in regard to