EXPLANATION OF TABLES 9 
TABLE 32.—COMMON LOGARITHMS OF NUMBERS. 
This table and the explanation thereto, have been prepared and copyrighted by Lyman M. 
Kells, Willis F. Kern, and James R. Bland, who have supplied them to the Hydrographic Office for 
use in its publications. Neither the table nor any new features embodied therein, may be reproduced 
in any form without the permission of the copyright owners. 
Additional examples in the use of logarithms are contained in Appendix III of Part J of this 
publication. 
Introduction.— The power L to which a given number b must be raised to produce a number N is 
called the logarithm of N to the base b. ‘This relation expressed in symbols is 
bL=N. 
It appears at once that 6 must not be unity and it must not be negative. In the following 
set of tables, 10 is used as base. 
Characteristic and mantissa.—The common logarithm of any real, positive number may be 
written as an integer, positive or negative, plus a positive decimal fraction. The integral part is 
called the characteristic and the decimal part the mantissa. The characteristic may be written by 
using the following rules: 
Rue 1: The characteristic of the common logarithm of a number greater than 1 is obtained by 
subtracting 1 from the number of digits to the left of the decimal point. 
For example, 68.30 has two digits to the left of its decimal point; hence its characteristic is 
2—1=1. Similarly for 6830, the characteristic is 4—1=8, for 7.864 it is 1—1=0, and for 5846300 
it is 6. 
Rue 2: The characteristic of the common logarithm of a positive number less than 1 is negative 
and its magnitude is obtained by adding 1 to the number of zeros immediately following the decimal 
joint. 
p If the characteristic of a number is —n (n positive), it should be written in the form (10—n) —10. 
To obtain directly the logarithm of a number less than 1, subtract from 9 the number of zeros immediately 
following the decimal point, and write the result before the mantissa and — 10 after it. 
For example, 0.000785 has three zeros immediately following the decimal point; hence its 
characteristic is — (3-++1)=—4, or6—10. Similarly for 0.0000587 the characteristic is —(4+1)=—5 
or 5—10, for 0.0287 it is —2 or 8—10, and for 0.684 it is —1 or 9—10. 
To find the mantissa—Special case——The mantissa, or decimal part of the logarithm of a 
number, depends only on the sequence of the digits and not on the position of the decimal point. 
Table 32 lists the mantissas, accurate to five decimal places, of the logarithms of all integers from 
1 to 10,000. 
The change in the mantissas of the logarithms is very slow. Consequently the first two digits 
of the mantissas have been omitted from a large percentage of entries. When these two digits are 
omitted from an entry, they always appear in the column containing the entry both slightly above 
it and also slightly below it. 
To find the mantissa of the logarithm of a number locate the first three digits of this number 
in the left-hand column headed No., and the fourth digit in the row at the top of the page. Then 
the mantissa of the given number containing four significant figures is in the row whose first three 
figures are the first three significant figures of the given number, and in the column headed by the 
fourth. Thus to find the logarithm of 76.64 find 766 in the column headed No., and follow the cor- 
responding row to the entry in the column headed by 4. This entry 88446 represents the mantissa 
required. The first two digits 88 of the mantissa were found in the same column with the considered 
entry but one space lower, and also in the same column, but seven spaces higher. 
Hence, we have 
4 log 76.64=1.88446. 
Interpolation—When a number contains a fifth significant figure, we find the logarithm cor- 
responding to the first four figures as above and then add an increment obtained by a process called 
interpolation. This process is based on the assumption that for relatively small changes in the nwmber 
N the changes in log N are proportional to the changes in N. The following example will serve to 
illustrate the process of interpolation. 
The expression tabular difference will be used frequently in what follows. The tabular differ- 
ence, when used in connection with a table, means the result of subtracting the lesser of two succes- 
sive entries from the greater. These differences have been computed in every case and tabulated 
in the columns headed ‘‘d’’. 
Examep.e: Find log 235.47. 
Solution.—We first find the logarithms in the following form and then compute the difference 
indicated: 
log ge la Seg 
log 235.47)" }10=? 18 (tabular difference). 
log 235.50 = 2.37199 
By the principle of proportional parts, we have 
Cd 
a 
a= = =12'6= ly). 
107 18 or d 0 12.6=13 (nearly) 
Adding 0.00013 to 2.37181, we obtain 
log 235.47=2.37194. 
