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EXPLANATION OF THE TABLES
5. Use of the Table. To use logarithms in computation we need a table arranged so as to enable us to find, with as little effort and time as possible, the logarithms of given numbers and, vice versa, to find numbers when their logarithms are known. Since the characteristics may be found by means of Rules I and II, p. vi, only mantissas are given. This is done in Table I. Most of the numbers in this table are irrational, and must be represented in the decimal system by approximations. A fiveplace table is one which gives the values correct to five places of decimals. PROBLEM 1. To find the logarithm of a given number. First, determine the characteristic, then look in the table for the mantissa.
To find the mantissa in the table when the given number (neglecting the decimal point) consists of four, or less, digits (exclusive of ciphers at the beginning or end), look in the column marked N for the first three digits and select the column headed by the fourth digit: the mantissa will be found at the intersection of this row and this column. Thus to find the logarithm of 72050, observe first (Rule I) that the characteristic is 4. To find the mantissa, fix attention on the digits 7205; find 720 in column N, and opposite it in column 5 is the desired mantissa, .85763; hence log 72050 4.85763. The mantissa of .007826 is found opposite 782 in column 6 and is .89354; hence log .0078267.89354 — 10.
6. Interpolation. If there are more than four significant figures in the given number, its mantissa is not printed in the table; but it can be found approximately by assuming that the mantissa varies as the number varies in the small interval not tabulated; while this assumption is not strictly correct, it is sufficiently accurate for use with this table.
Thus, to find the logarithm of 72054 we observe that log 72050 = 4.85763 and that log 72060 = 4.85769. Hence a change of 10 in the number causes a change of .00006 in the mantissa; we assume therefore that a change of 4 in the number will cause, approximately, a change of .4 × .00006 = .00002 (dropping the sixth place) in the mantissa; and we write log 72054 4.85763+.00002 = 4.85765.
The difference between two successive values printed in the table is called a tabular difference (.00006, above). The proportional part of this difference to be added to one of the tabular values is called the correction (.000002, above), and is found by multiplying the tabular difference by the appropriate fraction (.4, above). These proportional parts are usually written without the zeros, and are printed at the right-hand side of each page, to be used when mental multiplications seem uncertain.
Example 1. Find the logarithm of .0012647. Opposite 126 in column 4 find .10175; the tabular difference is 34 (zeros dropped); .7 × 34 is given in the margin as 24; this correction added gives .10199 as the mantissa of .0012647; hence log .00126477.10199 - 10. Example 2. Find the logarithm of 1.85743. Opposite 185 in column 7 find .26858; tabular difference 23; .48 × 23 is given in the margin as 10; this correction added gives .26868 as the mantissa of 1.85743; hence log 1.85743 0.26868.
7. Reverse Reading of the Table. PROBLEM 2. To find the number when its logarithm is known. First, fixing attention on the mantissa only, find from the table the number having this mantissa, then place the decimal point by means of the two following rules : *
RULE III. If the characteristic of the logarithm is positive (in which case the mantissa is not followed by 10), begin at the left, count digits one more than the characteristic, and place the decimal point to the right of the last digit counted.
RULE IV. If the characteristic is negative (in which case the mantissa will be preceded by a number n and followed by — 10†), prefix 9. ciphers, and place the decimal point to the left of these ciphers.
Example 1. Given log ∞ = 1.22737, to find x.
Since the mantissa is 22737, we look for 22 in the first column and to the right and below for 737, which we find in column 8 opposite 168. The number is therefore 1688. Since the characteristic is +1, we begin at the left, count 2 places, and place the point; hence x = 16.88.
Example 2. Given log x = 2.24912, to find ..
This mantissa is not found in the table; in such cases we interpolate as follows: select the mantissa in the table next less than the given mantissa, and write down the corresponding number; here, 1774; the tabular difference is 25; the actual difference (found by subtracting the mantissa of 1774 from the given mantissa) is 17; hence the proportionality factor is 17/25.68 or .7 (to the nearest tenth). Since moving the decimal point does not affect the mantissa, it follows that the digits in the required number are 17747 (to five places). The characteristic 2 directs to count 3 places from the left; hence x = 177.47.
RULE. In general, when the given mantissa is not found in the table, write down four digits of the number corresponding to the mantissa in the table next less than the given mantissa, determine a fifth figure by dividing the actual difference by the tabular difference, and locate the decimal point by means of the characteristic.
8. Illustrations of the Use of Logarithms in Computation.
Example 1. To find 832.43 × 302.43 X 16.725 .000178.
*Another convenient form of these rules is as follows: if the characteristic were zero, the decimal point would fall just after the first significant figure; move the decimal point one place to the right for each positive unit in the characteristic, one place to the left for each negative unit in the characteristic.
We might add the logarithms of the factors in the numerator and from this sum subtract the logarithm of the denominator; but we can shorten the operation by adding the negative of the logarithm of the denominator instead of subtracting the logarithm itself. The negative of the logarithm of a number (when written in convenient form for computation) is called the cologarithm of the number. We may find the negative of any number by subtracting it from zero, and it is convenient in logarithmic computation to write zero in the form 10.00000-10. Thus the negative of 2.17 is 7.83 10; the negative of 1.1432-10 is S.8568. Remembering that the cologarithm of a number is its negative we have the following rule:
To find the cologarithm of a number begin at the left of its logarithm (including the characteristic) and subtract each digit from 9, except the last,* which subtract from 10; if the logarithm has not 10 after the mantissa, write 10 after the result;
if the logarithm has 10 after the mantissa, do not write - 10 after the result. By this rule the cologarithm of a number can be read directly out of the table without taking the trouble to write down the logarithm. Attention must be given not to forget the characteristic. The use of the cologarithm is governed by the principle:
Adding the cologarithm is equivalent to subtracting the logarithm.
Returning to the computation of the given problem we should write:
We have no logarithms of negative numbers, but an inspection of this problem shows that the result will be negative and numerically the same as though all the factors were positive; hence we proceed as follows:
* If the logarithm ends in one or more ciphers, the last significant digit is to be under
9. The Slide Rule. A slide rule consists of two pieces of the shape of a ruler, one of which slides in grooves in the other; each is marked
(Fig. 1) in divisions (scale A and scale B) whose distances from one end are proportional to the logarithms of the numbers marked on them. It follows that the sum of two logarithms can be obtained by simply
sliding one rule along the other; thus if (see Fig. 2) the point marked 1 on scale B is set opposite the point marked 2.5 on scale A, the point on scale B marked 2 will be opposite the point on scale A marked 5, since log 2.5 + log 2 log 5. Likewise, opposite 3 (scale B) read 7.5 (scale A); opposite 2.5 (B) read 6.25 (A), i.e. 2.5 × 2.5 = 6.25.
Other multiplications can be performed in an analogous manner. Divisions can be performed by reversing the operation. Thus, if 4.5 (B) be set on 11.25 (A), then 1 (B) will be opposite 2.5 (A), as in Fig. 2.
Scales C and D are made just twice as large as scales A and B. It follows that the numbers marked on C and D are the square roots of the numbers marked opposite them on scales A and B.
For a description of more elaborate slide-rules, and full directions for use, see the catalogues of instrument makers.
The student should use a slide rule in checking results; practice may be had by checking many of the results of the following list of exercises.
EXPLANATION OF THE TABLES
10. Graphical Representation of Interpolation. In the process of interpolation, values are inserted as if the logarithm varied directly as the
number, between the two nearest values given in the table. Graphically, this means that the interpolation is made as if the curve y=logx consisted of a straight line segment.
If the values of x and y log x are plotted in the usual manner, the curve obtained is that shown in
Fig. 3. The values of x and y given in the table fall so close to each other on this figure that the interpolating line cannot be shown. But if the portion of the figure near x=2, y = .30103 be enlarged in
the ratio 1 to 10000 on the x-axis resulting figure is as shown in Fig. 4. .30125; the point B shows x = 2.002,
and 1 to 1000 on the y-axis, the The point A shows x = 2.001, y = y = .30146; if we draw the straight line ANB, it is clear that the straight line differs from the true curve AMB, but the difference is very slight.
Thus, the value of y given by interpolation for x = 2.0015 is shown at N; it is y = .301355. The true value of log 2.0015, found from a higher place table is really .3013556; but either of these results would be written .30136, so that the error made in using the straight line ANB in place of the curve AMB does not affect the fifth place of decimals.