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68. In the system of logarithms in common use, the base is 10; so that to construct a table of these, we have to solve the equations 10-1, 10x=2, 10x=3, &c.
We have at once the logarithms of the powers of 10, thus:
log. 100 log. 10-2 log. 10-2, (Art. 62.)
1=0, (Art. 64.) 10=1, (Art. 66.)
log. 1000=log. 103-3 log. 10—3,
log. 10000=log. 10*—4 log. 10—4,
log. 0.01 log.
log. 0.0001 log.
That is, the logarithm of a number composed of the figure 1 and ciphers, is equal to the number of places by which the figure 1 is removed from the units place, and is positive when this figure is to the left, and negative when it is to the right of the unit's place.
69. It follows from this, that if a number is
between 1 and 10, its log. is between 0 and 1,
1 66 2,
66 10 66
66 4, &c.
and if between 0.1 and 1, its log. is between 1 and
0.01 66 0.1,
❝ 0.0001 66 0.001, 66
70. The integral part of a logarithm is called its characteristic, and sometimes its index.
From the preceding table we perceive that if a number is between 1 and 10, that is if it contains only one integral figure, its logarithm is between 0 and 1 and consists of the characteristic 0 and a decimal fraction. If a number is between 10 and 100, it contains two integral figures, and its logarithm consists of the characteristic 1 and a decimal fraction; and, generally, a number between 10" and 10"+1 contains n integral figures, and its logarithm has the characteristic n.
Again, if a number is between 0.1 and 1, its logarithm is between —1 and 0, and is negative; thus, log. 0.3456=-0.46143. But this mode of expressing negative logarithms is never employed ; it is found more convenient to regard the logarithm of a decimal fraction as composed of two parts, a negative characteristic and a positive decimal fraction. According to this method we shall have log. 0.3456——1+.53857, which is equivalent to -0.46143. In like manner we shall have log. 0.03456=-2+.53857; and, gene
rally, a number being between and (that is, between 10- and 10-n+1,) its logarithm will have the characteristic -N. For the sake of conciseness, the characteristic is placed next the decimal with the minus sign over it; thus, log. 0.3456=1.53857, log. 0.03456-2.53857, which must not be mistaken for -1.53857,
From these considerations we derive the following rule:
The characteristic of the logarithm of a number is equal to the number of places by which the first significant figure of that number is removed from the unit's place, and is positive when this figure is to the left, negative when it is to the right, and zero when it is in the unit's place.
71. Let b represent any number whose first significant figure is in the unit's place; its logarithm will consist of the characteristic 0 and a decimal fraction, which represent by d. Then we shall have
log. 10 blog. 10 +log.b=1+d,
log. 100 b=log. 100+log. b=2+d,
and in general, log. 10"b= log. 10"+log. b=n+d,
log. 10 +log. b——1+d,
log. 100+log. b=-2+d,
log. 10"+log. b——n+d.
These equations prove, in a general manner, the preceding rule for the characteristic. We see also that the decimal part (d) of the b b
logarithm is the same for all the numbers b, 106, 10′′b,
10' 10 But multiplying or dividing a number by 10 does not change its significant figures, but merely removes the decimal point to the right or to the left. Hence we have another important property of common logarithms, that the same decimal part is common to the logarithms of all numbers composed of the same significant figures.
All this is exemplified in the following table:
12345 = 4.09149 1234.5 3.09149 123.45 = 2.09149
and in general, log.
72. On these properties depends the usual arrangement of the tables of common logarithms. None but whole numbers are placed in the table, and the characteristics of their logarithms are omitted. To find the logarithm of any number from the tables, we neglect the decimal point and take out the logarithm corresponding to the figures of which our number consists; we then prefix the charac
teristic according to the position of the decimal point, by the rule given above. On the other hand, in finding the number corresponding to a given logarithm, we regard only the decimal part of the logarithm and take out the corresponding figures from the column of numbers; the characteristic then determines the unit's place and the position of the decimal point.*
73. As the use of negative characteristics is apt to produce confusion in calculation by the mixture of positive and negative quantities, it is most generally avoided by adding 10 to the characteristic, which is thus rendered positive. The logarithm may then be used in this form throughout a calculation, at the end of which we must make the proper allowance, adding or subtracting 10, as the case may be. No error can arise from this, as it is hardly possible in a calculation to make a mistake of 10,000,000,000.
Thus, the log. of 0.3456 becomes-1+.53857+10=9.53857, and the log. of 0.03456 becomes -2+.53857+10=8.53857.
We have then this rule:
The characteristic of the logarithm of a decimal fraction is equal to 10 minus the number of places by which the first significant figure is removed from the unit's place.
If the negative characteristic were greater than 10, we should render it positive by adding 100. Thus, instead of 19.23471 we should use 81.23471, and reject 100 at the end of the calculation.
74. To find the product of two numbers.
This is effected by Art. 60; that is, by adding the logarithms.
To find the product of 4125 and 312. The process will be as follows:
log. 1287000 = 6.10957
* Particular directions for using tables are not here given, because every collection of tables is accompanied with the necessary explanations.
We find in the tables that the decimal part of the logarithm 6.10957 corresponds to the figures 1287; but the characteristic 6 shows that the first significant figure is removed 6 places from the unit's place. We must therefore add three ciphers, which gives 1287000 for the true product.
75. The use of logarithms is particularly convenient when the continued product of several numbers is sought.
To find the continued product of 41.87, 2.385, 9.4, 0.012 and 0.85.
log. 41.87 = 1.62190
log. 2.385 log. 9.4
log. 0.012 = 8.07918 (Art 73.)
Here, since the characteristics of log 0.012 and log. 0.85 are each 10 too great, that of the sum of the logarithms is 20 too great. Rejecting 20, then, the log. is 0.98112, which is the log. of 9.5746 the required product.
76. To find the quotient of two numbers.
This is effected by Art. 61; that is, by subtracting the log. of the divisor from the log. of the dividend.
To find the quotient of 5979.3 divided by 432.18.
log. 5979.3 3.77665
log. 13.835 = 1.14099
The required quotient is therefore 13.835.
To find the quotient of 597.93 divided by 43218.