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The characteristic is always one less than the places of integer figures in the number whose logarithm is taken.
Thus, in the first case, for numbers between 1 and 10, there is but one place of figures, and the characteristic is 0. For numbers between 10 and 100, there are two places of figures, and the characteristic is 1; and similarly for other numbers.
TABLE OF LOGARITHMS.
5. A table of logarithms, is a table in which are written the logarithms of all numbers between 1 and some other given number. The logarithms of all numbers between 1 and 10,000 are written in the annexed table.
6. The first column on the left of each page of the table, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed directly opposite them, and on the same horizontal line.
To find, from the table, the logarithm of any number.
7. If the number is less than 100, look on the first page of the table, along the column of numbers under N, until the number is found: the number directly opposite, in the column designated log, is the logarithm sought. Thus,
log 9 0.954243.
When the number is greater than 100, and less than 10,000. 8. Since the characteristic of every logarithm is less by unity than the places of integer figures in its corresponding number (Art. 4), its value is known by a simple inspection of the number whose logarithm is sought. Hence, it has not been deemed necessary to write the characteristics in the table. To obtain the decimal part of the logarithm, find, in the column of numbers, the first three figures of the given number. Then, pass across the page, along a horizontal line, into the columns marked 0, 1, 2, 3, 4, 5, &c., until you come to the column which is designated by the fourth figure of the given number: at this place there are four figures of the required logarithm. To the four figures so found, two figures taken from the column marked 0, are to be prefixed. If the four figures thus found, stand opposite to a row of six figures in the column marked 0, the two figures from this column, which
the four figures found are opposite a line of only four figures, you are then to ascend the column till you come to the line of six figures; the two figures, at the left hand, are to be prefixed, and then the decimal part of the logarithm is obtained; to which prefix the characteristic, and you have the entire logarithm sought. For example,
In several of the columns, designated 0, 1, 2, 3, 4, &c., small dots are found. When the logarithm falls at such places, a cipher must be written for each of the dots, and the two figures, from the column 0, which are to be prefixed, are then found in the horizontal line directly below.
the two dots being changed into two ciphers, and the 34 to be taken from the column 0, is found in the horizontal line directly below.
The two figures from the column 0, must also be taken from the horizontal line below, if any dots shall have been passed over, in passing along the horizontal line: thus,
the 49 from the column 0, being taken from the line 310.
When the number exceeds 10,000, or is expressed by five or more figures.
9. Consider all the figures, after the fourth from the left hand, as ciphers. Find from the table the logarithm of the first four figures, and to it prefix a characteristic less by unity than all the places of figures in the given number. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the figures that have been considered as ciphers: then cut off from the right hand as many places for decimals as there are figures in the multiplier, and add the product so obtained to the first logarithm, and the sum will be the logarithm sought.
Let it be required, for example, to find the logarithm of
log 672800 5.827886
the characteristic being 5, since there are six places of figures.
being multiplied by 87, the figures regarded as ciphers, gives for a product 5655; then pointing off two decimal places, we obtain 56.55 for the number to be added.
In adding the proportional number, we omit the decimal part; but when the decimal part exceeds 5 tenths, as in the case above, its value is nearer unity than 0; in which case, we augment by one, the figure on the left of the decimal point.
10. This method of finding the logarithms of numbers which exceed four places of figures, does not give the exact logarithm; for, it supposes that the logarithms are proportional to their corresponding numbers, which is not rigorously
To explain the reason of the above method, let us take the logarithm of 672900, a number greater than 672800 by 100. We then have,
In this proportion the first term 100 is the difference between two numbers, one of which is greater and the other less than the given number; and the second term 65 is the difference of their logarithms, or tabular difference.
The third term 87 is the difference between the given number and the less number 672800; and hence the fourth term 56.55 is the difference of their logarithms. This difference therefore, added to the logarithm of the less number, will give that of the greater, nearly.
Had there been three figures of the given number treated as ciphers, the first term would have been 1000; had there been four, it would have been 10000, &c. Therefore, the reason of the rule, for the use of the column of differences, is manifest.
To find the logarithm of a decimal number.
and a decimal, such as 36.78, it may be put under the form 3878. But since a fraction is equal to the quotient obtained by dividing the numerator by the denominator, its logarithm will be equal to the logarithm of the numerator minus the logarithm of the denominator. Therefore,
log log 3678-log 100=3.565612-2=1.565612 from which we see, that a mixed number may be treated as though it were entire, except in fixing the value of the characteristic, which is always one less than the number of the integer figures.
12. The logarithm of a decimal fraction is also readily found. For,
log 0.8=log=log 8-1=-1+log 8. But,
which is positive and less than 1. Therefore,
in which, however, the minus sign belongs only to the characteristic. Hence it appears, that the logarithm of tenths is the same as the logarithm of the corresponding whole number, excepting, that the characteristic instead of being 0, is-1.
If the fraction were of the form 0.06 it might be written; taking the logarithms, we have,
loglog 06-2=-2+log 06 -2.778151 in which the minus sign, as before, belongs only to the characteristic. If the decimal were 0.006 its logarithm would be the same as before, excepting the characteristic, which would be-3. It is, indeed, evident, that the negative characteristic will always be one greater than the number of ciphers between the decimal point and the first significant figure. Therefore, the logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to the decimal part of its logarithm a negative characteristic greater by unity than the number of ciphers between the decimal point ana the first significant figure.
That we may not, for a moment, suppose the negative sigu to belong to the whole logarithm, when in fact it belongs only to the characteristic, we place the sign above the characte ristic, thus,
To find in the table, the number answering to a given logarithm. 13. Search in the columns of logarithms for the decimal part of the given logarithm, and if it can be exactly found, set down the corresponding number. Then, if the characteristic of the given logarithm is positive, point off from the left of the number found, one more place for whole numbers than there are units in the characteristic of the given logarithm, and treat the figures to the right as decimals.
If the characteristic of the given logarithm is 0, there will be one place of whole numbers; if it is-1, the number will be entirely decimal; if it is -2, there will be one cipher between the decimal point and the first significant figure; if it is 3, there will be two, &c
The number whose logarithm is 1.492481, is found at page 5, and is 31.08.
But when the decimal part of the logarithm cannot be exactly found in the table, take the number answering to the nearest less logarithm; take also from the table the corresponding difference in the column D. Then, subtract this less logarithm from the given logarithm, and having annexed any number of ciphers to the remainder, divide it by the difference taken from the column D, and annex the quotient to the number answering to the less logarithm: this gives the required number, nearly. This rule, like that for finding the logarithm of a number when the places of figures exceed four, supposes the numbers to be proportional to their corresponding logarithms.
1. Find the number answering to the logarithm 1.532708.