Taking 10 for the base of the tables, we have,

The logarithms of numbers between 1 and 10 are expressed by decimal fractions; of those between 10 and 100, by unity, plus a fraction; of those between 100 and 1000, by 2 plus a fraction, &c.

(107.) The logarithm of any number, N, may be found by reducing the equation (10) = N, N representing the given number. Thus, in a former article, we found the value of x in the equation (10) = 2, to be 0,30107, which is the logarithm of 2. This fraction is a little too large, and by continuing the operation further, we should have obtained, 0,3010300, which is exact for seven places of decimals. In like manner, by reducing the equation (10)* = 3, we find x', or the logarithm of 3, equal to 0,4771213. In the same way we might find the logarithm of any number, but as the process is exceedingly tedious, mathematicians have devised more expeditious methods, which cannot here le made known.

(108). It is only necessary to calculate directly the logarithms of prime numbers, as the logarithms of all others may be derived from these. Thus, if we would find the logarithm of 4, we have only to double that of 2; for, taking the equation (10) = 2, and squaring both members, we have, (10) = 4; or taking the same equation, and cubing both members, we have (10) = 8; which shows, that twice the log. of 2 1.4, and three times the log. of 2 = 1.8. Again; if we would find the log. of any number, as 30, we have only to add together the logarithms of its prime factors, 30 = 3 × 5 × 2, and we have,

=

(10)* = 3, (10)" = 5, and (10)*"' = 2.

Now, if we multiply these three equations together, term by term, observing the rule for the exponents, in the first members, we shall have,

(10)*+*+*" = 30, or 1.3 +1.5 +1.2 = 1.30.

Tables of logarithms have been calculated for numbers extending from 1 to 100,000. For a description of their arrangement, and the manner of using them, the student is referred

to the introductions usually accompanying these tables, in which all necessary information is given.

(109.) We proceed now to show how numerical calculations are performed by means of logarithms.

Multiplication by Logarithms.

Let a be the base of a system of logarithms, and let y and y' be two numbers, the product of which is required; we shall have (106), a* = y, a" = y';

α

and multiplying these equations together, we have,

a+"=yy', or l.y + l.y' = l.yy' ;

which shows, that the sum of the logarithms of two numbers is equal to the logarithm of their product.

Hence, to multiply by logarithms, we are to add together the logarithms of two factors, and find in the table the number corresponding to their sum.

Division by Logarithms.

(110.) Suppose we have, as before, a* = y, a*' = y'. Dividing the first of these equations by the second, and observing that this division is effected in the first member, by subtracting the exponent a' from x, we have,

from which we see, that the difference of the logarithms of two numbers, is equal to the logarithm of their quotient. Hence, division is performed in logarithms by subtracting the logarithm of the divisor from the logarithm of the dividend, and finding the number, in the tables, which corresponds to the difference.

Involution by Logarithms.

(111.) If we take the equation a* = y, and raise both members to the 2d, 3d, 4th, and nth powers, we shall have (38), = y3, a12 = y1, a** = y" ; from which we see, that

a2* = y2,

A32

=

2 l.y = l.y2, 3 l.y = 1.y3, 41.y = l.y3, nl.y = l.y"; and hence, to involve a number by logarithms, we multiply the logarithm of the number, by the number denoting the

degree of the power, and find, in the tables, the number corresponding to the product.

Evolution by Logarithms.

(112.) Let ay: then if we extract the 2d, 3d, 4th, and nth roots of both members of this equation, we shall have, (45,)

3

4

n

a3 =vy, a3=vy, aỗ =√y, añ = vy;

or, fly = 1. Vy, ‡ ly = 1. Vy, ‡ Ly = 1. Vy,

1 1.y = 1.vy;

n

from which it follows, that the logarithm of any root of a number is the logarithm of the number itself divided by the index of the root.

Reduction of Exponential Equations by Logarithms. (113.) If we take the general equation bc, we shall have, by supposing the logarithms of both members known, 1.6* =l.c.

It has been shown, that the logarithm of any power of a number, is equal to the logarithm of the number itself, multiplied by the exponent of the power (111); hence, l.b* = x l.b, and therefore we have,

dividing by l.b,

x l.b

l.c; l.c

x= 1.6

For a numerical example; let it be required to find the value of x from the equation 8*= 128, we shall have,

(114.) Having shown how numerical calculations are performed by means of logarithms, we will now consider, more particularly, the properties of the common system of logarithms, formed upon the base 10.

We have seen, that in this system, the log. of 1=0, 1.10 1, 1.1002, 1.10003, 1.100004, and that all intermediate numbers have for their logarithms, the same whole numbers, plus a fraction. The integral part of a logarithm can, therefore, be known from the number of places of figures contained in the number to which it belongs. For

numbers less than 10, it is 0; for those greater than 10, and less than 100, it is 1; for those greater than 100, and less than 1000, it is 2, &c., being always one less than the number of figures expressing the given number. The integral part of a logarithm is called the characteristic of the logarithm.

(115.) It may also be shown, that the decimal part of the logarithms of numbers, one of which is 10, 100, 1000, &c. times greater or less than the other, is the same; for the logarithms of 10, 100, 1000, &c. are whole numbers, and being added or subtracted, would only alter the value of the characteristic, and would not affect the decimal part.

For example; 1.28 × 10, or 1.280 = 1.28 +1.10,

For the same reason, the logarithm of any number of figures, whether whole numbers or decimals, will be the same, with the exception of the characteristic, which will always be one less than the number of figures in the integral part of the number. For example; taking the number 654685, and dividing it successively by 10, and for each division subtracting from its logarithm the logarithm of 10, we have,

From this example, it also appears, that the characteristic of a decimal fraction is negative. It is always determined by the place which the first significant figure of the decimal occupies, counting from the decimal point towards the right. If it occupy the first place, the characteristic is -1; if the second, it is -2, &c., being obtained by subtracting the logarithm of 10, 100, 1000, &c. from zero.

(116.) The logarithms of all fractional numbers are negative, since they result from the subtraction of a greater loga,

rithm from a less. To obtain the logarithm of, for example, we have (110), 1.2 1.3 1.,

or,

=

Negative logarithms would be inconvenient in calculation, and to avoid their use we reduce the fraction to a decimal, the logarithm of which is positive, except the characteristic.

To effect this reduction, we have only to consider the numerator multiplied by 10, 100, or 1000, as may be necessary, which is done by adding 1, 2, or 3 to its characteristic, and then subtract the logarithm of the denominator. The number of units necessary to be added to the characteristic of the numerator, will determine the characteristic of the resulting logarithm, which must always, as has been shown (115), be negative.

Thus, to obtain the logarithm of, we multiply the numerator by 10, which gives for the numerator, 20 tenth parts of a unit, which divided by 3, will give a quotient in tenth parts. To effect this division by logarithms, we subtract the log. of 3 from the log. of 20:

1. 20= 1,3010300
1. 30,4771213

0,8239087

and since this difference is the log. of 10th parts, we must af fix to it 1 for a characteristic; or, which is the same thing, considering this difference the log. of a whole number, subtract the log. of 10 from it, to effect the division by 10, which gives 1 for the characteristic.

In practice, it is usual to perform the subtraction, as though the logarithm of the numerator were the greater, till we come to the characteristic, we then take the excess of the characteristic of the log. of the denominator over that of the numerator, and affect it with the negative sign, for the characteristic of the resulting logarithm. The result is the same as by the preceding method.

For example; let it be required to find the quotient of 5, divided by 73 in decimals, by means of logarithms.