But com. log 10 = 1, and Nap. log 10 = 2.30258508, as found above. Hence,

195. As one of the most important uses of logarithms is to facilitate the performance of multiplication, division, involution, and evolution, when the numbers are large, according to (178-181), it is necessary to have at hand a table containing the logarithms of numbers. Such a table of common logarithms is usually found in treatises on trigonometry and on surveying, or in a separate volume of tables.* These tables usually contain the common logarithms of numbers from 1 to 10000, with provision for ascertaining therefrom the logarithms of other numbers with sufficient accuracy for practical purposes. Four pages of such a table will be found at the close of this volume.

196. Prob.-To find the logarithm of a number from the table.

SOLUTION.-The logarithm of any number from 1 to 100 inclusive can be taken directly from the first page of the table. Thus log 20.301030, and log 21 = 1.322219.†

To find the logarithm of any number from 100 to 999 inclusive, look for the number in the column headed N, and opposite the number in the first column at the right is the mantissa of the logarithm. The characteristic is known by (185). Thus log 182 = 2.260071; log 135 = 2.130334.

To find the logarithm of any number represented by 4 figures, find the first 3 left-hand figures in column N, and opposite this at the right in the column which has the fourth figure at its head, will be found the last four figures of the mantissa. The other two figures of the mantissa will be found in the 0 column, oppo

* Mathematicians and practical computers generally use more complete and extended tables than those found in connection with such elementary treatises. The common tables give five places of decimals in the mantissa. Those in connection with this series give six. Callet's tables edited by Hasler are standard eight-place logarithms. Vega's tables are among the best. Dr. Bremiker's edition, translated by Prof. Fischer, is a favorite. Köhler's edition of Vega's contains Gaussian logarithms. Vega's tables are seven-place. Ten-place logarithms are necessary for the more accurate astronomical calculations. Prof. J. Mills Peirce, of Harvard, has recently issued an elegant little folio edition of tables containing among other things a table of three-place logarithms which is very convenient for most uses.

+ This page is really unnecessary, since nothing can be found from it which cannot be found with equal case from the succeeding part of the table. Thus, the mantissa of log 2 is the same as the mantissa of log 200; and the mantissa of log 21 is the same as that of log 210.

site the first three figures of the number or just above, unless heavy dots have been passed or reached in running across the page to the right, in which case the first two figures of the mantissa will be found in the 0 column just below the number. The places of the heavy dots must be supplied with O's. The characteristic is determined by (185). Thus log 1316=3.119256; log 2042=3.310056; log 1868 3.271377.

To find the logarithm of a number represented by more than 4 figures. Let it be required to find the logarithm of 1934261. Finding the mantissa corresponding to the first four figures (1934) as before, we find it to be .286456. Now in the same horizontal line and in the column marked D, we find 225, which is called the Tabular Difference. This is the difference between the logarithms of two consecutive numbers at this point in the table. Thus 225 (millionths) is the difference between the logarithms of 1934 and 1935, or, as we are using it, between the logarithms of 1934000 and 1935000, which differences are the same. Now, assuming that, if an increase of 1000 in the number makes an increase of 225 (millionths) in the logarithm, an increase of 261 in the number will make an increase of, or, .261, of 225 (millionths) in the logarithm,* we have .261 × 225 (millionths) = 59 (millionths), omitting lower orders, as the amount to be added to the logarithm of 1934000 to produce the logarithm of 1934261. Adding this and writing the characteristic (185) we have log 1934261 = 6.286515. In like manner the logarithm of any other number expressed by more than four figures may be found.

197. SCH.-As the mantissa of a mixed integral and decimal fractional number, or of a number entirely decimal fractional, is the same as that of an integral number expressed by the same figures (184), we can find the mantissa of the logarithm of such a number as if the number were wholly integral, and determine the characteristic by (185).

198. Prob.-To find the number corresponding to a given logarithm.

SOLUTION.-Let it be required to find the number corresponding to the logarithm 4.234567. Looking in the table for the next less mantissa, we find .234517, the number corresponding to which is 1716 (no account being taken as to whether it is integral, fractional, or mixed; as in any case, the figures will be the same). Now, from the tabular difference, in column D, we find that an increase of 253 (millionths) upon this logarithm, would make an increase of 1 in the number, making it 1717. But the given logarithm is only 50 greater than the logarithm of 1716; hence, it is assumed (though only approximately correct) that the increase of the number is of 1, or .1976 +. This added (the figures annexed) to 1716, gives 17161976 +. The characteristic of the given logarithm being 4, the number lies between the 4th and 5th powers of 10, and hence has 5 integral places. . 4.234567 = log 17161.976 +. In like manner the number corresponding to any logarithm can be found.

* This assumption, though not strictly correct, is sufficiently accurate for all ordinary purposes.

199. Prop.-The Napierian base is 2.718281828.

DEM.-Let e represent the base of the Napierian system. Then by (190)

com. loge Nap. log e:: .43429448 : 1.

But the logarithm of the base of a system, taken in that system is 1, since a1= a. Hence, Nap. log e= 1, and com. log e = .43429448. Now finding fron a table of common logarithms the number corresponding to the logarithm .43429448, we have e 2.718281828.

EXAMPLES.

1. If 3 were the base of a system of logarithms, what would be the logarithm of 81 Of 729 If 5 were the base, of what number would 3 be the logarithm? Of what 2? Of what 4?

Of

2. If 2 were the base, what would be the logarithm of ? Of ?

?

3. If 16 were the base, of what number would .5 be the logarithm? Of what .25?

4. In the common system we find that log 156=2.193125. Show that this signifies that 10188888–156.

5. Log 1955 3.291147. To what power does this indicate that 10 is to be raised, and what root extracted to make 1955 ?

6. Find from the table at the close of the volume what root of what power of 10 equals 2598.

7. Multiply 1482 by 136 by means of logarithms, using the table at the close of the volume. (See 178.)

8. Perform the following operations by means of logarithms: 1168 × 1879; 2769 187; 15.13 × 1.3476; 257.16 18.5134; .1266.1413; .11257 × .00126; (1278.6); (112.37)3.

9. Perform the following operations by means of logarithms: √2 to 5 places of decimals; 5 to 3 places of decimals; 2341564273 to two places of decimals; √3015618 to 4 places of decimals.

10. Perform the following operations by means of logarithms: V.01234 to 4 places of decimals; √.03125 to 5 places of decimals. V.0002137 to 5 places of decimals.

SUG'S.-Log .01234-2.091315. Now to divide this by 3, we have to remember that the characteristic alone is negative, i. e. that 2.091315= == - 2+.091315, or

L

-1.908685, which is all negative. Dividing this by 3, we have -.636228, or 0-.636228-1.363772. But a more convenient way to effect the division is to write 2.091315=3+1.091315, and dividing the latter by three we obtain 1.363772, in which the characteristic alone is negative, thus conforming to the tables.

To divide 13.341652 by 4, we write for 13.341652, -16+3.341652, and dividing the latter obtain 4.835413.

11. Divide as above 11.348256 by 3; 17.135421 by 5; 1.341263 by 6.

12. Given the following to compute x by logarithms:

201.56: 134.201 :: 18.654:x;
x: 234.008::15.738: 200.56;

2350.64.212::1.1123: x;
123:x:: 2.01: .03.

to express the equivalent operations

y = √(a − x) (a + x)+(1 + x).

in logarithms.

-log (1+x)].

14. Given y=x3 (1—x2) to express the equivalent operations in

SUG'S.-Write y = log (a + x) + log (a− x). Then differentiating, we have

a + x
2mxdx

a

x

Or differentiating without factoring, we have dy :

d(a2 — x2)* a2-x2

When reduced the results are the same, but the former is usually

the more elegant method.

16. Differentiate the following: y = log (1-x); y = log ax;

*This form signifies that a2-x2 is to be differentiated. The operation is only indicated, not performed.

SUG's.-Remember that log 3 3 log x; and also that log /1 + x = log (1 + x).

17. Find from the table at the close of the volume that Nap. log 1564 7.3550018. Find in like manner the Napierian logarithms of 5, 120, and 2154372.

18. Knowing that the Napierian logarithm of 22 is 3.0910425, how would you find the common logarithm of 23 from the logarithmic series (192)?

19. The common logarithm of 25 is 1.39794. What is the modulus, and what the base of a system which makes the logarithm of 25 2.14285 ?

QUERY.-How do you see at a glance that the required base is a little less than 5?

SECTION V.

SUCCESSIVE DIFFERENTIATION, AND DIFFERENTIAL

COEFFICIENTS.

200. Prop.-Differentials, though infinitesimals, are not necessarily equal to each other.

DEM. Thus, let y=2x3. Then dy=6x2dx. Now, for all finite values of x, dy is an infinitesimal, since no finite number of times the infinitesimal da can make a finite quantity, and dy is 6x2 times da. But for x=1, dy is 6 times dx; for x=2, dy is 24 times da; for x=3, dy is 54 times dx.

201. COR.-When y=f(x), dy is generally a variable, and hence can be differentiated as any other variable.

202. NOTATION.-The differential of dy is written day, and read "second differential of y." The differential of d'y is written d3y, and read "third differential of y," etc. The superiors 2 and 3 in such cases are not of the nature of exponents, as the d is not a symbol of number.

203. In differentiating y=f(x) successively, it is customary to regard de as constant. This is conceiving x to change (grow) by equal infinitesimal increments, and thence ascertaining how y varies. In general, y will not vary by equal increments when x does, as appears from the demonstration above.