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Thus 397940 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.
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 twice 10, by three times 10, becomes successively 234.8 23.48 2.348. The decimal part of the logarithms of these last three numbers, will be the •same, the characteristic being one less each time that we divide by 10 or remove the decimal point one place to the left. The characteristic of the first, 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 10'.
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
of 482 is 2
.482 is -1
.0482 is - 2 of
.00482 is - 3 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 logarithms of proper fractions ought to be negative, appears from the fact, that since a fraction expresses the quotient of the numerator divided by the denominator, applying the rule
for division by logarithms, the greater logarithm would have to be subtracted from the lesser and the remainder would of course be negative.
From the above principles are derived the following rules :
1. To find the logarithm of a number consisting of significant figures with any number of ciphers annexed, find the logarithm of the significant figures, and make the characteristic one less than the number of figures in the given number including the ciphers.
2. To find the logarithm of a decimal or mixed number, consider the number as entire ; find the decimal part of its logarithm, and make the characteristic one less than the number of figures in the entire part of the given number.
4. To find the logarithm of a decimal number having ciphers at the left ; look for the logarithm of the significant figures, and make the characteristic negative* and one more than the number of ciphers at the left of the given decimal.
The logarithm of 3266000
is 6.514016 of 114.1
is 2.057286 of .001684
is 3.226342 51. We proceed now to the method of determining the logarithm of a number beyond the limits of the table. This method is by a simple calculation from the logarithms of numbers which the table contains, and depends upon the fact that the difference of any two numbers bears the same proportion to the difference of their logarithms, that the difference of two other numbers does to the difference of their logarithms, which is nearly true.
Take two numbers in the table differing from each other by 100 as the numbers 843700 and 813800 and a third number 843742 differing from the first of these by 42. The loga
* It is customary to write the negative sign over the characteristic, thus, 2.1756348. It affects the characteristic alone and not the decimal part of the logarithm.
rithm of the first number 843700 is given by the tables and
is 5.926188 The logarithm of the second number 843800 is 5.926240
Thier difference is
*52 which may be found by subtraction, but to save this trouble the subtraction is performed and the difference is written in the column marked D, the last of each page in the table. Then diff. of numbers diff of logs.
diff. of num.
diff. of logs.
.000051 x 42
100 adding this to the logarithm of 843700 which is 5.926188
the sum, rejecting the last two places 42 which go beyond the usual number is
5.926209 which is the logarithm of 843742.
Had the first two numbers differed by 1000 instead of 100 the divisor in the value of x would have been 1000 and the quotient would have extended three places beyond the usual.
The inaccuracy of this method increases with the number of additional figures beyond four, in the number the logarithm of which is to be found.
From the above process may be observed the following rule:
To find the logarithm of a number beyond the limits of the table. Enter the table with the first four figures of the given number, and find the corresponding logarithm. From the column marked i take out the number opposite to this logarithm, and multiply it by the remaining figures of the proposed number, reject from the product as many figures to the right as there are in the multiplier, and add the rest of the product to the logarithm already found.
* The remainder is 52, but if the decimals had been carried beyond six places in the tables, it would have been 51.
Required the logarithm of 739245.
the number in column D is
Product, 2655 From this product reject as many figures to the right as are contained in the multiplier, that is two in this case, and add the rest to the logarithm before found, namely
868762 The sum is
868789* which is the decimal part of the log. of 739245 required. Prefixing the proper characteristic, we have 5.868789.
TO FIND THE NUMBER CORRESPONDING TO ANY GIVEN
52. By referring to the proportion of art. 51, and putting the value of x for the fourth term we have diff. of num. diff. of logs. diff. of num.
diff. of logs.
* We add 1 for the buto rejected which is more than ..
Instead of the 42 being given and the 000021 required as before, the 000021 is now given and the 42 required.
The first term of the proportion is 100 or 1000, &c., and the second term is in the column marked D, to find the third term multiply the extremes and divide by the second term
000021 x 100 42
000051 Hence the following
To find the number corresponding to any given logarithm.
Seek for the decimal part of the given logarithm, and we shall be readily guided to it, or else to the logarithm very near it, by means of the leading figures which are separated in the table from the others, to attract the eye. If we find a logarithm exactly agreeing with that given, then the number, which the table shows us to belong to the logarithm found will be the required number. If however, as is most likely we do not find the proposed logarithm exactly, then we are to take out the number corresponding to the next less logarithm; this number will of course fall short of that required, but the deficiency may be supplied as follows. Subtract the tabular logarithm from the given one, annex ciphers to the remainder at pleasure, and divide it by that number in the column which is opposite to the tabular logarithm, and annex the quotient to the number already taken from the table.
N. B. Should there be a quotient figure without annexing a cipher to the dividend, this quotient figure must be added to the last figure of the number taken from the table. Should it be necessary to annex two ciphers before obtaining a quotient figure, a cipher must be placed in the quotient and annexed with the figures that come after to the number taken from the table.
The logarithm next greater than that given may be taken from the tables, and the latter subtracted from the former in