Here is evidently equal to zero. (Algebra, Art. 17.) Hence in every system, log. of 10.

45. Suppose now the system be the common system; b will be equal to 10. If we substitute for n all possible numbers successively, we shall have a series of equations like the following,

In the first is the common logarithm of 1, in the second of 2, in the third of 3, &c. If I be made the unknown quantity, and these equations be successively resolved, we shall have the common logarithms of all numbers.* * If now a table be formed having the series of natural numbers, 1, 2, 3, 4, &c., in one column, and their logarithms calculated as above placed in a second column against them, this would be a table of logarithms. The tables in actual use do not differ from such an one in principle, though some arrangements are adopted in them to avoid unnecessary repetitions.t

In the common system the logarithm of 10 is 1, the logarithm of 100 is 2; and the logarithms of all numbers between 10 and 100 are between 1 and 2, that is, they are 1 and a fraction. The logarithm of 1000 is 3, and the logarithms of all numbers between 100 and 1000 are between 2 and 3, that is, they are 2 and a fraction. In the same manner it may be shown that the logarithms of all numbers between 1000 and 10,000 are 3 and a fraction; of all numbers between 10,000 and 100,000, 4 and a fraction, and so on. The logarithms of most numbers, therefore, are mixed numbers. The fractional part is written in the tables; the whole number part, which is called the characteristic, is not written, nor is it necessary that it should be; for numbers between 10 and 100, or those composed of two figures, have 1 for a characteristic, as has just been

The method of resolving them is given in Art. 325, Alg.

+ For a more full exposition of the theory of logarithms, see Algebra, Art. 210, page 258, et seq.

seer;

numbers between 100 and 1000, or those containing three figures, have 2 for a characteristic; numbers containing four figures have 3 for a characteristic, and so on. Whence it appears that the characteristic is always 1 less than the number of digits in the number to which the logarithm belongs. So that if against any given number, the decimal part of its logarithm be found in the tables, the entire part or characteristic may be supplied by counting the figures in the given number, and making the characteristic, one less.

In proceeding to explain the tables, we will premise that the logarithms of several consecutive numbers, if the numbers be somewhat large, will differ so little as to have several of their first figures the same. Hence, by a proper arrangement of the tables, the first figures of the logarithm may be written but once for several numbers, provided all be designated to which they refer, and thus much repetition be avoided.

The manner in which this is accomplished will be shown in the

EXPLANATION OF THE TABLES.

PROBLEM I.

46. To find from the tables the logarithm of any given number. CASE I. When the number is between 100 and 10,000, if it be composed of three figures, find it in Table XXVI. at the end of the volume, and in the column at the left entitled No.; in the next column marked 0 at top and on the same horizontal line you will find the decimal part of the logarithm required. This contains five places.* If the given number contain fourt figures, find the first three of it in the column No. as before, and the fourth in one of the columns marked 0, 1, 2, 3, &c., at top; under the latter, and on the same horizontal line with the first three, you will find the decimal part of the logarithm sought.

N. B. The characteristic is always one less than the number of figures in the given number.

* In the tables of Callet seven places, the first three of which being the same for several numbers are not repeated, but must be understood before those which follow the number that has them expressed until you come again to seven places.

In the tables of Callet substitute the word five for four in the above rule, and the word four for three.

In the tables of Callet you find only the last four places, the first three to be prefixed to them must be taken from the numbers projecting to the left in the column marked 0 at top.

EXAMPLES.

1. Required the logarithm of 217.

In the column entitled No. on page 171 of Table XXVI. I find 217; in the next column marked 0 at top, and on the same horizontal line, I find 33646 for the decimal part of the logarithm required. The characteristic is 2, since 217 contains three figures, and the whole logarithm 2.33646. 2. Required the logarithm of 1122

On page 170 and in the column No. I find 112; in the column having the last figure 2 of the given number at top, and on the same horizontal line with the 112 before found, I find 04999 or with the proper characteristic 3.04999.*

3. Required the logarithm of 2188.

Ans. 3.34005.

47. We proceed now to show the use of logarithms in numerical calculations.

MULTIPLICATION.

Let b be the base of the system of logarithms, n any number, and its logarithm. Then by the definition

b' = n

Let n' be another number, and l' its logarithm, we have also

b'' =n'

Multiplying these two equations, member by member, and observing the rule for exponents in multiplication, which is to add them together, we have

= nn'

From this last expression, it appears that +' is the exponent of the power to which it is necessary to raise the base a, in order to produce the number nn'. But nn' is the product of n and n'. Hence the logarithm of the product is equal to the sum of the logarithms of the multiplier and multiplicand.

At the tops of the pages in the table will be found catch numbers for the eye, in turning over the leaves, to show what numbers and logarithms are contained on the page.

Multiply 2421 by 1613.

EXAMPLE.

The logarithm of 2421 is

The logarithm of 1613 is

3.38399
3.20763

The logarithm of 2421×1613 or 3905073 is* 6.59162, or the sum of the logarithms.

48. If in addition to the numbers n and n'above, we suppose a third number n" of which the logarithm is l' we shall have in a similar manner

Or, in general, the logarithm of a product of several factors is equal to the sum of the logarithms of those factors separately.

we have, observing the rule of division, to subtract the exponent of the divisor from that of the dividend in order to obtain that of the quotient.

Since l' is the exponent of the power to which it is necessary to raise b the base, in order to produce it follows that—l' is the loga

n

N

n'

rithm of i. e. the logarithm of the quotient is equal to the difference

As the number 3905073 is too large to be found in the tables, the method of finding its logarithm from the tables must be postponed till the explanation of such cases, further in advance.

Before explaining other operations by means of logarithms, we shall exhibit some principles derived from those just demonstrated.

50. The base of the common system being 10, the common logarithm of 10 is 1. (Art. 44.) Hence if any number be multiplied or divided by any number of times 10, the logarithm of the result will be equal to the logarithm of the given number increased or diminished by the same number of times 1. This 1 being an entire number, the decimal part of the logarithm of the given number will not be altered by this addition or diminution, but only the characteristic.

Thus 39794, which is the decimal part of the logarithm of 2500, is also of 25000, and of 250000, or of 250, or of 25.

The characteristics belonging to these different numbers are different. That of the log. of 2500 is 3; that of the log. of 25000 is 4; that of the log of 25 is 1. (See Art. 47.)

Any number is divided by a multiple of 10, by pointing off from the right as many places for decimals, as the divisor is times 10.

Thus 2348 divided by 10, by 10 twice, by 10 three times, becomes successively 234.8, 23.48, 2.348. The decimal part of the logarithms of these last three numbers, will be the same, viz. 37070, the characteristic being one less each time that we divide by 10 or remove the decimal point one place to the left. Because to divide by 10 it is necessary (see the last Art.) to subtract 1, which is the log. of 10, from the log. of the dividend. The characteristic of the first, 234.8, which is between 100 102 and 1000 = 103, is 2. The characteristic of the second is 1 ; and the characteristic of the last is 0, since 2.348 is less than 10, or 101. The decimal part of the logarithm of a number consisting of significant figures, either followed or preceded by ciphers, will be the same as if the ciphers were absent. Thus the decimal part of the logarithm of 482000 or of ⚫00482 is the same as the decimal part of the logarithm of 482. The following table illustrates the theory of the characteristic. The characteristic of the log. of 482000 is 5

From the above, it appears that the characteristic of the logarithm of a decimal fraction is negative; the decimal part of the same logarithm is, however, positive. The actual value of the whole logarithm will be therefore a negative quantity somewhat less than the characteristic. That the