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ITHMS are a series of numbers so contrived, that by them the work of multiplication is performed by addition, and that of division by subtraction.
If a series of numbers in arithmetical progression be placed as indices, or exponents, to a series of numbers in geometrical progression, the sum or difference of any two of the former, will answer to the product or quotient of the two corresponding terms of the latter. Thus,
0. 1. 2. 3. 4.
5. 6. 7. &c. arith. series, or indices. 1. 2. 4. 8. 16. 32. 64. 128. &c. geom. series.
Now 2 + 3 = 5.
also 7-3= 4.
Therefore the arithmetical series, or indices, have the same properties as logarithms; and these properties hold true, whatever may be the ratio of the geometrical series.
There may, therefore, be as many different systems of logarithms, as there can be taken different geometrical series, having unity for the first term. But the most con
venient system, is that in which the ratio of the geometrical series is 10; and this is the one in common use. Thus,
0. 1. 2. 3. 4. 5. &c. indices, or logar. 1. 10.. 100. 1000. 10000. 100000. &c. natural numbers.
In this system the log. of 1 is 0,* the log. of 10 is 1, the log. of 100 is 2, &c. Hence it is plain, that the log. of any number between 1 and 10, will be expressed by a decimal, the log. of any number between 10 and 100, by 1 and a decimal, the log. of any number between 100 and 1000, by 2 and a decimal, &c.
The numbers 0, 1, 2, 3, &c. that stand before the decimal part of logarithms, are called indices, and are always less by unity, than the number of figures in the integral part of the corresponding natural number.
The index of the logarithm of a number, consisting in whole, or in part of integers, is affirmative; but if the number be a decimal, the index is negative, and is marked by a negative sign (-) placed either before or above it. If the first significant figure of the decimal be adjacent to the decimal point, the index is -1, or ī; if there be one cipher between them, the index is -2, or 7; if there be two ciphers between them, the index is --3, or 3, &c.
The decimal parts of the logarithms of numbers, consisting of the same figures and in the same order, are the same, whether the number be integral, fractional, or mixed. This is illustrated as follows:
The method of finding logarithms in the tables, and of multiplying, dividing, &c. by them is contained in the following problems.
To find the logarithm of a given number.
If the given number consist of one or two figures only, find it in the column marked No. in the first page of the table, and against it in the next column, marked log. is the logarithm. Thus the log. of 7 will be found 0.81510, and the log. of 85 will be found 1.92942.
But if the given number be, either wholly or in part, decimal, the index must be changed accordingly. Observing that the index must always be one less, than the number of figures in the integral part of the given number; also, when the given number is wholly a decimal, the index is negative, and must be one more than the number of the cyphers between the decimal point and first significant figure on the left hand. Thus the log. of .7 is -1.84510, and the log. of .0085 is -3.92942.
If the given number consist of three figures, find it in one of the other pages of the table, in the column marked
No. and against it, in the next column, is the decimal part of the logarithm. The index must be placed before it agreeably to the above observation. Thus the log. of 421 is 2.62428, the log. of 4.21 is 0.62428, and the log. of .0421, is 2.62428.
If the given number consist of four figures, find the three left hand ones in the column marked No. as before, and the remaining or right hand figure at the top of the table; in the column under this figure, and against the other three, is the decimal part of the logarithm. Thus the log, of 5163 is 3.71290, and the log. of .6387 is -1.80530.
If the given number consist of five or six figures, find the logarithm of the four left hand figures as before; then take the difference between this logarithm and the next greater in the table. Multiply this difference by the remaining figure or figures of the given number, and cut off one or two figures to the right hand of the product, according as the multiplier consists of one or two figures; then add the remaining figure or figures of the product to the logarithm first taken out of the table, and the sum will be the logarithm required. Thus, let it be required to find the logarithm of 59686; then, Logarithm of 5968 is
77583 The next greater log. is
The natural number consisting of five integers, the index must be 4; therefore the log. of 59686 is 4.77587.
Again, let it be required to find the log. of .0131755;
Logarithm of 1317 is
1. Required the log. of
2. Required the log. of 3. Required the log. of 4. Required the log. of 5. Required the log. of
6. Required the log. of
Decimal part of the log.
As the given number is a decimal, and has one cipher between the decimal point and first significant figure, the index must be -2; therefore the log. of .0131755 is -2.11977.
Because 17.6 is nearer 18 than 17.