16. How would you transform a system of logarithms from base 8 to base 4?

17. Find log25 8, having given log10 2.

18. Find a from the equation 8=100, given log 2.

19. Solve 5*-1= 4, given log 2.

23. Solve 23× × 63x−5 – 32(x−1) × 25*, given log 2 and log 3.

=

24. Given log 2, find log 128, log 125, and log 2500.

26. Find log sin 45°, log tan 60°, log sec 30°; given log 2 and log 3.

27. Find log 3, log 16 and log 450.

Given log 18

=

1.255272, log 25 = 1.397940.

28. Given log 2 and log 3; find the number of integral digits in

320 × 5:5
211

29. Given log 2 and log 3; find the value of

30. Given log 2 and log 3; find the logarithms of the mean proportional between

(1.25) and (1.25)

(*00016)6

1000

CHAPTER XIV.

LOGARITHMIC TABLES.

123. Tables of Logarithms. From the properties of logarithms treated of in the last chapter will be seen the advantage of having a compilation of the logarithms of the numbers 1, 2, 3, 4, &c. to as high a number as possible. Such a compilation is what is called a table" of logarithms of the "natural" numbers. We have also other equally useful tables of logarithms, as will appear. Good tables are easily and cheaply procurable, and moreover usually contain a description of the manner in which they are to be used. Hence it will be needless to explain in this treatise the mere manipulation of a copy of them, if the student furnish himself with one; and more needless if he do not. We shall confine ourselves therefore to explaining the simpler and more useful principles of the subject.

124. Advantage of Briggs', or the Common Logarithms. A table of common logarithms has the great advantage of being very much less bulky than a table computed to any other base than 10; and on two accounts :

(1.) The characteristics of the common logarithms of numbers may all be determined by inspection of the numbers; and the registering of them is therefore dispensed with.

(2.) Having registered the mantissæ of certain numbers, those of all others having the same significant figures are the same, and therefore do not require a separate statement.

125. Principle of Proportional Parts. The common tables give the logarithms of whole numbers from 1 to

100,000; which may be made available for all numbers or decimal fractions whatever with not more than five significant figures. By the "principle of proportional parts," they may also be made available for numbers with more than five significant figures, not with perfect accuracy, but with sufficient accuracy for ordinary practical purposes. As applying to numbers, the principle may be stated as follows: The change in the logarithm of a number is approximately proportional to the change in the number itself, if it be small in comparison with the magnitude of the number. As applying to angles and their functions it will be hereafter enunciated. It is of course capable of proof; which, however, is beyond the scope of this treatise.

126. To find the logarithm of a number of more than five significant digits.

The characteristic is seen on inspection; and we need therefore only concern ourselves with finding the mantissa.

Place the decimal point of the number after the fifth significant digit of it. The mantissa will not be affected by this proceeding.

Let N+n represent the number as thus altered, N being its integral, and n its decimal part.

Also let d = log (N+1)-log N,

Now d is known from the tables; for log N and log (N+1) are consecutive logarithms therein; and d is required.

By the principle of proportional parts,

therefore

and is known.

since

therefore

And

d: 8:1: n;

d = nd,

log (N+n)—log N = 8,

log (N+n) = log N+nd,

and is also known; and its mantissa is that which is required.

127. Given a logarithm which is not in the tables, to find the number whose logarithm it is.

With the notation of the previous article, evidently N+n is first required, while N, d, & are known from the tables and the given logarithm. Now, as before, d:d :: 1:n;

which gives a number with the same significant digits as the one required. The decimal point may then be placed in accordance with the value of the given characteristic.

128. In that part of the tables in which the numbers N and N+1 occur, the value of d would be called the difference for 1; d therefore is the difference for n; but is usually called the "proportional part" for n.

129. It will be best to exemplify practically the preceding propositions at once.

(1.) Find the logarithm of the number 349-54572.

Wherefore, to find the proportional part for 572, we have, writing d for it, by the principle of proportional parts,

OBS.-When cutting off some of the digits in a decimal fraction, add unity to the last digit retained, if the first of those cut off be 5 or more than 5.

(2.) Find the number whose logarithm is 4-8026755.

Finding in the tables the two mantissæ between which the given mantissa lies, we have

Wherefore, if n be the decimal fraction corresponding to this proportional part, we have

130. Tables of Proportional Parts.-The multiplication and division of the preceding examples being somewhat tedious, the best tables of logarithms replace them by means of what are called "tables of proportional parts" for each "difference." These tables are easily constructed, and for the differences we have employed would stand thus :

in which the parts 14, 27, &c. are 1 of 136, 2 of 136, &c.; and 7, 14, &c. are 1 of 69, 2 of 69, &c. The differences and proportional parts as given above are apparently integers; but they are really decimal fractions of seven places, the points and ciphers being omitted for convenience.