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loga N.

log, NP

.

27

10822

Let it be required to find loga No.
Assume N a“, and therefore x =
We have NP (am)?

= apm, and hence, by definition,

= = p loga N. 0.Ě.D. Ex. 1.-Given log 2 = 3010300, and log 3 = .4771213, find the logarithms of 18, 15, 125, 675. Log 18 = 34

log (2 x 32) = log 2 + 2 log 3 = 3010300 + 2('4771213) = 1·2552726.

Log 15 = log (3 x 40) log 3 + log 10 log 2 = 4771213 + 1 •3010300 1.1760913.

1

2 = 0 - 3 x 3010300 ..9030900 1 + (1 - 59030900) 1 + •0969100, or, as usually written, = 1:0969100.

33 Log6.75 = log

= 3 log 3 – 2 log 2 = 3( 4771213)

4 – 20-3010300) = .8293039. .

(2-4)+ x Ex. 2. Find the logarithm of

x (375)*

having

(2-43)* * (-032)
given log 2 and log 3.
We have-
Log N = log 2-4 + 4 log 375 - 5 log 2:43

- (- }) log .032.
1
23 x 3

3

35 1 25
+ 4 log

log
10
28

3 103
1 (3 log 2 + log 3 - log 10) + 4 (log 3 – 3 log 2)
- 5 (5 log 3 - 2 log 10) + } (5 log 2 - 3 log 10)
(1 - 12 + x) log 2 + (! + 4 - 25) log 3

+ (-1 + 10 - 1) log 10
– 5,3 x 3010300 4 x .4771213 + x 1
2.6590983 9.7809867 + 8.5 3.9400850
= 2 + (4 - 3.9400850) = 4 0599150.

log

5 log 10%

+

8

.81

Ex. IV. 1. Find the logarithm to base 4 of the following numbers: 16, 64, 2, 25, .0625, 8. 2. Find the value of log 32, log 25, log :729.

V5 3. Given log 2 = -3010300, and log 3 = .4771213, find the logarithms of 12, 36, 45, 75, 04, 3-75, 6, 07).

4. Given log 20763 = 4.3172901, what is the logarithm of 2.0763, 2076-3, .020763, .0020763 ?

5. Write down the characteristics of the common logarithms of 29.6, -25402, 0034, 6176.003.

6. Given log 20.912 = 1.3203956, what numbers correspond to the following logarithms :-23203956, 6-3203956, 1.3203956, 4.3203956 ?

7. Given log 20.713 = 1.3162430, and log 20714 = 3162640, find log 2071457.

8. Given log 3.4937 = •5432856, and log 3.4938 •5432980, find the number whose logarithm is 3.5432930. 9. Given log 1.05 = .0211893, log 2:7 = 1.4313638, log

(2.7) 135 = 2.1303338, find the value of log

x 513-5

(1:05) 10. Given log 18 = 1.2552725, and log 2.4 = 3802112, find the value of log .00135. 11. What are the characteristics of log 1167, and log 1965? 12. Having log 2 = '3010300, and log 3 = -4771213, find a when 18* 125.

CHAPTER VI.

THE USE OF TABLES.

25. Tables have been formed of the logarithms of all numbers from 1 to 100,000, and we shall now show how they are

practically used. We shall not enter here upon the method of forming the tables themselves.

The following is a specimen of the way in which the logarithms of numbers are usually tabulated :

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5

11

16

22

27

32

38

43

49

៩ { 54 sil Thus, if the number consist of four figures only, we have simply to copy out the figures in the column headed 0, prefix a decimal point, and the proper characteristic.

Ex. Log 7991 3.9026011, log 7.995 •9028185. When we speak of a number consisting of four figures only, we include such numbers as ·003654, •07682, &c., the number of zeros immediately following the decimal points not being counted.

Thus, log .07997 2.9029271

log .007992 = 3.9026555. When the number contains five figures, as, for instance, 79936, we look along the line containing the first four figures viz., 7993—of the number until the

eye

the column headed 6, the fifth figure. We then take the first three figures of the column headed 0, and affix the four figures of the column headed 6 in the horizontal line of the first four figures of the number.

Thus, log 79936 = 4.9027424

log .079927 = 2.9026935. It will be seen from the portion of the logarithmic table above extracted, that when the first three figures of the logarithm—viz., 902—have been once printed, they are not

rests upon

repeated, but must be understood to belong to every four. figures in each column, until they are superseded by higher figures, as 903. When, however, this change is intended to be made at any place not at the commencement of a horizontal row, the first of the four figures corresponding to the change is usually printed either in different type, or, as above, with a bar over it. Thus we have above 0031, indicating that from this point we must prefix 903 instead of 902.

Thus, log 79.986 = 1.9030140,

log .0079987 = 3.9030194. 26. To find the logarithm of a number not contained in the tables.

Ex. Find the logarithm of 799.1635.

Since* the mantissa of the number 79916.35 is the same as the mantissa of the given number, and that the first five figures are contained in the tables, we may proceed as follows

(1.) Take out from the tables the mantissa corresponding to the number 79916. This is .9026337.

(2.) Take out the mantissa of the next higher number in the tables-viz., 79917. This is :9026392.

(3.) Find the difference between these mantissæ. This is called the tabular difference, being the difference of the mantissæ for a difference of unity in the numbers. We find tab. diff. •0000055, which we call D.

(4.) Then assuming that small differences in numbers are proportional to the differences of the corresponding logarithms, we find the difference for •35 •35 x .0000055 •0000019, retaining only 7 figures. This is often called d.

(5.) Now adding this value of d to the mantissa for the number 79916, we get the mantissa corresponding to the number 79916:35.

(6.) Lastly, prefix to this mantissa the proper characteristic. The whole operation may stand thusM. of log 79916 .9026337....

(1). M. of log 79917 = .9026392 Tabular difference or D. •0000055 * Thus log 79916-35 = log (100 x 799·1635) = 2 + log 799 1635..

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Hence, difference for 35 or d
-35 x .0000055 :.0000019...

(2). Hence, adding (1.) and (2.)—

M. of log 79916:35 .9026356.

log 799·1635 2.9026356. or better thus, omitting the useless ciphers

M. of log 79916 .9026337
M. of log 79917 .9026392

55
Hence, d = .35 x 55

19 .. M. of log 79916:35 •9026356, as before. In the next article we shall show how the required difference may be obtained by inspection from the tables.

27. Proportional parts.

We saw in the example just worked that the tab. diff. (omitting the useless ciphers) is 55, and if we examine the table in Art. 25, we shall find the difference between the mantissæ of any two consecutive numbers there to be 54 or 55—generally 54. The number 54 is therefore placed in a separate column at the right of the table, and headed D.

The student will understand that the tab. diff. changes from time to time, and is not always 54 or 55.

Now assuming as in (4.) of the last article, we haveDiff. for •1 54 x •1 = 5 Diff. for •6 54 x 6 32 •2 54 x .2 - 11

54 x 57 38 •3 54 x 3 = 16

•8 54 x 8 43 •4 54 x .4 22

.9 54 x .9 49 •5 54 x 5 - 27

We find therefore the numbers 5, 11, 16, 22, 27, 32, 38, 43, 49 placed in a horizontal row at the bottom marked P, in the columns respectively headed 1, 2, 3, 4, &c.

Hence, if we require the difference for (say) 7, we take out the number 38 from the horizontal row marked P, instead of being at the trouble to find it by actual computation.

The following example will illustrate how we proceed when we require the difference for a decimal containing more than one decimal figure. No explanation is needed.

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