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154. Our common logarithms are formed from the Logarithms of the Napierian System by multiplying each of the latter by a common multiplier called The Modulus of the Common System.

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This modulus is, in accordance with the conclusion of Art. 152,

log, 10°

That is, if 7 and No be the logarithms of the same number in the common and Napierian systems respectively,

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and so the modulus of the common system is 43429448.

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156. The following are simple examples of the method of applying the principles explained in this Chapter.

Ex. (1). Given log 2 = 3010300 and log 3 = 4771213, find the logarithms of 64, 81 and 96.

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.. log 96 = 5 log 2 + log 3 = 1·5051500+ ·4771213=1.9822713.

Ex. (2). Given log 56989700, find the logarithm of (6.25).

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(1) Given log 2 = 3010300, find log 128, log 125 and log 2500.

(2) Given log 2 = 3010300 and log 7 = 8450980, find the logarithms of 50, 005 and 196.

(3) Given log 2 = 3010300, and log 3 = 4771213, find the logarithms of 6, 27, 54 and 576,

(4) Given log 2 = 3010300, log 3=4771213, log 7 =8450980, find log 60, log 03, log 1.05, and log 0000432.

260

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(5) Given log 23010300, log 18= 1.2552725 and

log 21 = 1.3222193, find log 00075 and log 31.5.

(6) Given log 5 = '6989700, find the logarithms of 2, ·064, and

14

590

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(7) Given log 2 = 3010300, find the logarithms of 5, .125 and

1

15

(8) What are the logarithms of 01, 1 and 100 to the base 10? What to the base '01 ?

(9) What is the characteristic of log 1593, (1) to base 10, (2) to base 12?

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(10) Given 8, and x = 3y, find x and y.

(11) Given log 4 = 6020600, log 1.04 0170333:

=

(a) Find the logarithms of 2, 25, 83-2, (625)

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(b) How many digits are there in the integral part of (1.04)6000?

(c) In how many years will a sum of money double itself, at 4 per cent. compound interest, payable yearly?

(12) Given log 25 1.3979400, log 1030128372:

1 100

(a) Find the logarithms of 5, 4, 51·5, (·064)1oo

(6) How many digits are there in the integral part of (1·03)600 ? (c) In how many years will a sum of money double itself at 3 per cent. compound interest, payable yearly?

(13) Having given log 3 = ·4771213, log 7 = ·8450980,

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CHAPTER XIII.

ON TRIGONOMETRICAL AND LOGARITHMIC TABLES.

157. We shall give in this Chapter a short description of the Tables which have been constructed for the purpose of facilitating Trigonometrical calculations.

The methods by which these Tables are formed do not fall within the range of this treatise: we have merely to explain how they are applied to the solution of such simple examples as we shall hereafter give.

We shall arrange our remarks in the following order:

I. On Tables of Logarithms of Numbers.

II. On Tables of Trigonometrical Ratios.

On Tables of Logarithms of Trigonometrical Ratios.

I. On Tables of Logarithms of Numbers.

158. These Tables are arranged so as to give the mantissæ of the logarithms of the natural numbers from 1 to 10000, that is of numbers containing from one to five digits.

We shall now shew how by aid of these tables, first, to find the logarithm of any given number, and, secondly, how to determine the number which corresponds to a given logarithm.

159. When a number is given to find its logarithm.

When the given number has not more than five digits we can take its logarithm at once from the tables.

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