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1 Now, A X 3
A X7 = .0000046, and
= .00000106 = .0000011, as we only 10
10 take in seven places of decimals. Hence, log. 2856837 = 64558798 + .0000046 + .0000011 = 6.4558855.
We have now given a full explanation of the principles on which Logarithmic Tables are calculated, so far as that explanation is possible in a purely elementary treatise, and have exemplified those principles in the case of the ordinary tables which (practically) give the logarithms of numbers from 1 up to 10,000,000, to seven places of decimals. The student may ask, what would be done if a case occur in which we have numbers exceeding 10,000,000 : The answer to the question is the following :If the number were, for instance, 97536982, and if the calculation demanded so much accuracy that we could not consider this as equal to 97536980, then a more refined set of tables would be necessary; in point of fact, however, for all ordinary calculations, the degree of accuracy which the common tables allow of is sufficient.
We now proceed to explain the practical method of using the tables :
THE USE OF A TABLE OF LOGARITHMS.
For the purpose of explanation, the following is printed from p. 78 of Hülsse's edition of Vega’s Logarithms.
(1). To find the Logarithm, when the Number is given in the Tables. (a) To find the mantissa.
Suppose we want to 'find the logarithm of 46017. The number N. 46000 at the top of the page will direct us to the page on which we shall find 46017; the number 4601, in the column marked N, will give us the line in which we shall find what we want. Pass your eye along the line 4601, until it comes to column 7, there we find 9183. You will observe that there is 662 written in column 0; this is to be written before every one of the numbers under the other columns, and is only written once to render the tables more compact. Write this in front of 9183, which we before found, and we obtain 6629183. This is the mantissa of the logarithm of 46017; it is, therefore, a decimal, and must be written •6629183.
So again, to find mantissa of logarithm of 46035. look down col. N for 4603, look along the line 4603 till you come to col. 5, when you find 0881· before this prefis 663, and .6630881 is the mantissa of logarithm of 46035.
Again, to find the mantissa of logarithm of 46028, look for 4602 in col. N; along this line, in col. 8, you find *0221; the asterisk in front of this 0221 shows that we must add 1 to the 662, and thus we obtain for the mantissa of this logarithm 6630221.
If you examine these cases, you will find that they suggest the following rule:“Place your pen between the first and second FIGURE (NOT cipher), and count one for each figure or cipher, until you come to the decimal point, the number this gives will be the characteristic; if you count to the right, the characteristic is positive, if to the left, the characteristic is negative. Thus, in finding log. 4·6017, if you place your pen between the first figure (4), and second (6), it falls on the decimal point, in this case, therefore, there is no characteristic. Next, in the case of log. 4601·7. place your pen between (4) 4,601.7 and (6), and count the characteristic is 3; and as you count to the right, it is | 123 plus 3. Next, in the case log. 4601700, here the decimal point falls behind the last 4 601700 cipher. Hence, counting as before, we have and the characteristic is plus | 123456, 6. Again, in the case, log. 00046017; the first figure is, as before, 4. Hence counting, ⚫0004, 6017, we have "but here we count to the left, so that the characteristic is nega4321 4 6017, and the characteristic is 1. 1 |
tive or 4. Again, in the case, log. 4601. we have
Instead of writing log. ·03046017 =4·6629182, this is frequently written 6·6629182. 46629182, which clearly equals
- 10. and write
10 although it is not written
To explain this, observe that 4.6629182 means 66629182 10, or 6.6629182 10. It is usual to omit the 6-6629182, no experienced calculator would forget the down, but as this tract is intended for beginners, we shall never omit the - 10, but as it may suit our purpose write log. 000460174·6629182 or 6·6629182 - 10. To find log. 46. Since 46 46·000 and the table gives mantissa log. 46000 = ⚫6627576.
.. log. 46
Hence, to find the logarithm of any number gixen in the table, first find the mantissa, and then prefix to it the characteristic, in the manner above explained.
N.B.-The student must thoroughly master the above before proceeding further. He must get a table of logarithms (of which there are many by Hutton, Callet, Babbage, &c., all as perfect as those of Vega's, to which reference has been madethere is one also published in Chambers's Educational Course, which is cheaper than most others)—and will work out many examples, such as the following:-Find the logarithms of
(2). To find the Logarithm of a Number not given in the Tables.
The rule for the characteristic is the same as given above. For finding the mantissa we proceed as follows:-The student will observe that each logarithm on p. 284 differs from the one before it by 94 or 95. Call this 95 and construct a table of proportional parts as before explained; this is printed in the column marked PP. We then proceed as follows:-To find log. *0460267.
In like manner the student may find the logarithms of
(3). To find the Number corresponding to a given Logarithm.
It very rarely happens that the logarithm is exactly to be found in the tables. If it is, the only difficulty we have to contend with in such a case is that of fixing the decimal point. For instance, find the number corresponding to the logarithm 3.6629089. At the top of the page we have L 662; this will direct us to the page on which the logarithm will be found; then, looking in the other part of the table we find 9089, in the column 6 of the line marked 4601 of column N. .. the number corresponding to the mantissa •6629089 is 46016. To fix upon the position of the decimal point, we must modify the rule previously given: place the pen between the first and second figure and count off as many figures as there are units in the characteristic,-to the right if the characteristic is positive, to the left if negative, and if there are not figures enough add or prefix as 4:601.6 many ciphers as necessary; thus in the present case the number correspond123 ing to logarithm 3:6629089 is 4601.6, similarly, that corresponding to 3-6629089 is 00:46016.
(4). To find the Number corresponding to a given Logarithm, which does not exactly occur in the Tables. We proceed as follows:
Find the number corresponding to the logarithm 2 6629319.
The Logarithms 6629277 and 6629372 are in the table; the number, therefore, will be between 46018 and 46019. It will therefore be the former, with something added To find this “something" we proceed as follows :—
.. logarithm of 460·1843 is 2.6629319.
(5). To find the Arithmetical complement of the Logarithm of a Number. N.B. If x is any number whatever, then the ar. comp. of x 10 -X. Now 10
3.7568274 = 6.2431726.
10 -2.3907526 = 11.6092474.
If you examine these, you will find that the subtraction is performed by subtracting the last figure (to the right hand) from 10, and each of the rest from 9; in fact, to take the first case, we should proceed as follows: 4 from 10 leaves 6, and carry 1. Then 1+7= 8; take 8 from 10, leaves 2, and carry 1; but taking 8 from 10 is of course the same thing as taking 7 from 9, and so on. The student may, perhaps, think this very obvious, but he will do well not to despise it.
Hence, to find the ar. comp. of the logarithm of a number, find the logarithm and subtract it from 10, in the manner above explained.
(6.) To find the product of several numbers by means of a Table of Logarithms.
log. N = log. x + log. y + log. z +, &c. . . . Hence, find the logarithm of each number, add them together this gives the logarithm of the quotient find the number corresponding to this logarithm, and we have the product itself.
N.B.-In any example of this kind, never use a negative characteristic, such as 2-8802418. but 8.8802418 - 10 as above; by doing so, there is nothing but straight forward addition to be performed until the end, when — 10 can be easily struck off the characteristic of the sum.
(7.) To divide one number by another by means of a Table of Logarithms.
Hence, "To log numerator, add ar. comp. logarithm of denominator, and subtract 10 from the sum this gives logarithm of quotient.-Find number corresponding to this logarithm, and the number is the quotient required."
(9). To find any power of a Number, we have seen that if N = an log. N = n log. a.
Hence, if we multiply log. of the number by the index, we obtain the logarithm of the power of the given number; and finding the number corresponding, we obtain the power itself.