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(11) Given log:9=a, log25=b, log:7=c; find the logs to base 10 of numbers 1 to 7 inclusive.

(12) How many positive integers are there whose logarithms to base 2 have 5 for a characteristic?

(13) If a be an integer, how many positive integers are there whose logs to base a have 10 for their characteristic ? (14) Given log 2 and log 7, find the eleventh root of (39-2).

log 1.9485=•289688. (15) Prove that 7 log 18 + 6 log $ + 5 log * +log 3:=log 3. (16) Prove that

2 log a +2 log a2 +2 log a3 ... +2 log an=n(n+1) log a. (17) Prove that log.b.log,a=1; and that logab.log.c. log.a=l. (18) Prove that log.r=log.b. log.c. log.d...log,r.

(19) Given that the integral part of (3.456)100000 contains 53856 digits, find log 345.6 correct to five places of decimals.

(20) Given that the integral part of (3.981)100000 contains sixty thousand digits, find log 39810 correct to five places of decimals.

(21) If the number of births in a year be is of the population at the beginning of the year, and the number of deaths ob, find in what time the population will be doubled.

Given log 2, log 3, and that log 241=2-3820170.
(22) Prove that
log 8 +log (8-a) - log b-log c=2 log

8 (s-a)

bc (23) Prove that

log (a? + x?) + log (a + x) +log (a - x)=log (a* - 24). (24) Prove that

log sin 4A=log 4 + log sin A + log cos A + log cos 24.

CHAPTER XV.

ON THE USE OF MATHEMATICAL TABLES.

213. The Logarithms referred to in this chapter, and in future throughout the book, are Common Logarithms.

214. Books of Mathematical Tables usually give an explanation of their own contents, but there are some points common to all such Tables which we proceed to explain.

215. The student will be supposed to have access to a book containing the following:

(i) A list of the logarithms of all whole numbers from 1 to 99999, calculated to seven significant figures;

(ii) A list of the numerical values, calculated to seven significant figures, of the Trigonometrical Ratios of all angles, between 0° and 90°, which differ by l';

(iii) A list of the logarithms of these Ratios calculated to seven significant figures.

These will be found in Chambers' Mathematical Tables.

216. We have said that logarithms are in general incommensurable numbers. Their values can therefore only be given approximately.

If the value of any number is given to seven significant figures, then the error (i.e. the difference between the given value and the exact value of the number) is less than a millionth part of the number.

Example. 3:141592 is the value of a correct to seven significant figures. The error is less than •000001 ; for a is less than 3:141593, and greater than 3:141592.

The ratio of .000001 to 3:141592 is equal to 1: 3141592. The ratio of .000001 to a is less than this; i.e. much less than the ratio of one to one million.

217. An actual measurement of any kind must be made with the greatest care, with the most accurate instruments, by the most skilful observers, if it is to attain to anything like the accuracy represented by seven significant figures.'

Therefore the value of any quantity given correct to seven significant figures' is exact for all practical purposes.

218. We are given in the Tables the logarithms of all numbers from 1 to 99999; that is, of any number having five significant figures.

A Table consisting of the logarithms of all numbers from 1 to 9999999 (i.e. of any number having seven significant figures) would be a hundred times as large.

219. There is however a rule by which, if we are given a complete list of the logarithms of numbers having five significant figures, we can find the logarithms of numbers having six or seven significant figures.

Example. Suppose we require the logarithm of 4-804213.
From the Tables we find

log 4.8042=6816211, i.e. 4.8042=10-6816211...,
log 4.8043=-6816301,

4.8043=10.6816301.... The number 4.804213 lies between the two numbers 4:8042, 4•8043 whose logarithms are found in the Tables, so that the required logarithm must lie between the two given logarithms.

Therefore we suppose that
log 4.804213=6816211+d, i.e. 4.804213=10.6816211...+d.

220. The RULE is as follows. The differences between three numbers are proportional to the corresponding differences between the logarithms of those numbers, provided that the differences between the numbers are small compared with the numbers.

Example. Thus in the above example 4.8042, 4.8043 and 4:804213 are three numbers ; '6816211, .6816301 and 6816211+d are their three logs

The difference between the first and second numbers is .0001.

The difference between the first and third numbers is 000013.

The difference between the logarithms of the first and second numbers is .000009.

The difference between the logarithms of the first and third numbers is d. By the Rule these differences are in proportion

.: *0001 : 000013=#000009 : d,

100 : 13=.000009 : d; whence

d=·00000117...,
.. log 4.804213=-6816211+ •00000117...
= .68162227=•6816223 (to seven figures).

or

221. We shall refer to the above rule as the Rule of Proportional Differences.

It is often called also "The Principle of Proportional Parts.

222. In Art. 197 we said that numbers are not proportional to their Logarithms. Hence the differences of numbers and the corresponding differences of their logarithms cannot be exactly in proportion. The rule is however true for all practical purposes. The proof of the rule belongs to a higher part of the subject than the present.

223. In the above example we said that

6.68162227 = 6.6816223; and for this reason. We are retaining only seven significaut figures in the decimal part of the logarithm.

If we put 6.6816222 for 6.68162227 the error' is greater than 00000007.

If we put 6.6816223 for 6.68162227 the error' is less than •00000003.

Thus the second error is less than the first.

In such a case, 1 must be added to the last digit which is retained, when the first digit which is neglected is 5 or greater than 5.

224. We give two more specimen examples.
Example 1. Find the logarithm of •004804213.

We first find as before, by the rule of proportional differences, that

log 4:804213=-6816223
.. log •004804213=3.6816223.

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