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Naperian system. And the modulus of any system is the logarithm of the base of the Naperian system, taken in that system of which it is the modulus.
47. We shall confine our attention now to the common system, the base of which is 10.
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 shown; 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.
48. To find from the tables the logarithm of any given number.
CASE. I. When the number is less than 100, find it in the small table on the first page of the tables, in one of the columns entitled N. The number found on its right upon the same horizontal line in the next column marked Log. will be its logarithm.
CASE. II. When the number is between 100 and 10,000, if it be composed of three figures, find it in the table commencing on page 2, and in the column entitled N; 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 six places, the first two 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 six places. If the given number contains four figures, find the first three of it in the column N 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 four places of the decimal part of the logarithm sought; the first two places to be prefixed to these, are to be taken from the column marked at top. It sometimes happens that in the columns which follow that marked 0, the partial logarithms to which the projecting figures in the 0 column refer are exhausted before you come to the right of the table. This will be indicated by the four places in the column where it occurs, expressing a less number than those in the preceding columns. The two figures to prefix found in the 0 column are then those below instead of above the horizontal line in which you are. In fact some of the first places of the four in the last column, to which the projecting figures above refer, will in this case be ciphers, and these ciphers are represented by dots to call the attention; so that in passing back from the column which has your fourth figure at top, if you pass over dots on
the same horizontal line take the two projecting figures below in the column marked 0 instead of those above.
N. B. The characteristic is always one less than the number of figures in the given number.
Required the logarithm of 217.
In the column entitled N on page 3 of the Tables, I find 217; in the next column marked 0 at top, and on the same horizontal line I find 6460; projecting to the left a little above I find 33, which I prefix to the 6460 and have 336460 for the decimal part of the logarithm required. The characteristic is 2, and the whole logarithm 2.336460.
Required the logarithm of 1122.
On page 2 and in the column N, 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 9993; projecting to the left in the 0 column and a little above the horizontal line in which I am, I find 04 which prefixed gives me 049993 or with the proper characteristic 3.049993.
Required the logarithm of 2188.
In the column N, I find 218, on the same horizontal line with which and under 8, I find 0047, the dots being ciphers; I pass to the column 0, but instead of taking the projecting figures above, I take those below, namely 34, and I have 340047 or with the characteristic 3.340047 the logarithm required.
In the same manner the logarithm of 1178 is found to be
49. We proceed now to show the use of logarithms in numerical calculations.
Let a be the base of the system of logarithms, n any number, and its logarithm. Then by the definition,
Let n' be another number, and l' its logarithm, we have also
a" = n'
Multiplying these two equations, member by member, and observing the rule for exponents in multiplication, which is to add them together, we have
altl' = nn'
From this last expression, it appears that +l' 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.
Multiply 2421 by 1613.
The logarithm of 2421 is
The logarithm of 2421×1613 or 3905073 is* 6.591629, or the sum of the logarithms.
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
and so on.
a2+"+" = nn'n'"'
Or in general the logarithm of a product of several factors is equal to the sum of the logarithms of those factors seperately.
* 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 to the explanation of such cases, in advance.
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 is the exponent of the power to which it is necessary to raise a the base, in order to produce it follows that is the logarithm of l l'
i. e. the logarithms of the
quotient is equal to the difference between the logarithms of the divisor and dividend.
Divide 3905073 by 2421
The logarithm of 3905073 is 6.591629
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.