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36. In an orchard of fruit trees, of them bear apples,
pears, 5 plumbs, and 50 of them cherries : how many trees are there in all ?
Ans. 600. 37. A can do a piece of work alone in 10 days, and B in 13 ; if both be set about it together, in what time will it be finished ?
Ans. 51 days. 38. A, B and C are to share 100000l. in the proportion
1 / and respectively ; but C's part being lost by his death, it is required to divide the whole sum properly.between the other two.
Ans. A's part is 57142 i 7, and B's 4285732".
END OF ARITHMETIC
LOGARITHMS are numbers, so contrived, and adapt ed to other numbers, that the sums and differences of the former shall correspond to, and shew, the products and quotients of the latter.
Or, logarithms are the numerical exponents of ratios; or a series of numbers in arithmetical progression, answering to another series of numbers in geometrical progression,
So, 1, 2, 3, 4, 5, 6, indices, or logarithms.
I, 2, 4, 8, 16, 32, 64, geometric progression.
6, indices, or logar.
ind. or log.
og I, 2, 3, 42 5,
Where it is evident, that the same indices serve equally for any geometric series ; and consequently there may be an endless variety of systems of logarithms to the same common numbers, by only changing the second term 2, 3,
or 10, &c. of the geometrical series of whole numbers ; and by interpolation the whole system of numbers may be made to enter the geometric series, and receive their pro. portional logarithmns, whether integers or decimals.
It is also apparent from the nature of these series, that if any two indices be added together, their sum will be the index of that number, which is equal to the product of the two terms in the geometric progression, to which those indices belong. Thus, the indices 2 and 3, being added together, make 5 ; and the numbers-4 and 8, or the terins corresponding to those indices, being multiplied together, make 32, which is the number answering to the
In like manner, if any one index be subtracted from an, other, the difference will be the index of that number, which is equal to the quotient of the two terms, to which those indices belong. Thus, the index 6 minus the index A=2; and the terms corresponding to those indices are 64 and 16, whose quotient =4; which is the number an, swering to the index 2.
For the same reason, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thus, the index or logarithm of 4, in the above series, is 2 ; and if this number be multiplied by 3, the product will be 56 which is the logarithm of 64, or the third power
4. And if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm of that root. Thus, the index or logarithm of 64 is 6; and if this number be divided by 2, the quotient will be =3 ; which is the logarithm of 8, or the square root
The logarithms most convenient for practice are such, as are adapted to a geometrical series, increasing in a tenfold proportion, as in the last of the above forms ; and are
those, those, which are to be found at present, in most of the common tables of logarithms.
The distinguishing mark of this sýstem of logarithms is, that the index or logarithm of 10 is 1 ; that of 100 is 2 ; that of 1000 is 3, &c. And, in decimals, the logarithm of 'I is -1 ; that of 'or is -2 ; that of .001 is 3, &c. the logarithm of í being o in every system.
Whence it follows, that the logarithm of any number between 1 and 10 must be o and some fractional parts ; and that of a number between 10 and 100, I and some 'fractional parts ; and so on, for any other number what
And since the integral part of a logarithm, thus readily found, shews the highest place of the corresponding number, it is called the index, or characteristic, and is commonly omitted in the tables ; being left to be supplied by the person, who uses them, as occasion requires.
Another definition of logarithms is, that the logarithm of any number is the index of that power of some other number, which is equal to the given number. So if there be N=r", then n is the log. of N; where n may be either positive or negative, or nothing, and the root r any number whatever, according to the diflerent systems of logarithms.
When n is = 0, then N is 1, whatever the value of q is; which shews, that the logarithm of 1 is always o, in every system of logarithms.
When n is =1, then N is =r; so that the radix r is always that number, whose logarithm is i in every system.
When the radix r is =2.718281828459, &c. the indices n are the hyperbolic or Napier's logarithm of the numbers N; so that n is always the hyperbolic logarithm of the number N or 2-713, &c.l.
But when the radix r is = 10, then the index n becomes the common or Briggs' logarithm of the number N ; so
that the common logarithm of any number 10" or N is na the index of that power of 10, which is equal to the said number. Thus, 100, being the second power of 10, will have 2 for its logarithm ; and 1000, being the third power of 10, will have 3 for its logarithm : hence also, if 50 be
3 =10769897, then is 1'69897 the common logarithm of 50. And, in general, the following decuple series of terms, viz. 104, 103, 10", 10", 10°, 10-, 10-, 10-3, 104,
or 10000, 1000, 100, 10,
'ol, '001, '0001,
3, 2, I, O, I, 2, -3, -4, for their logarithms, respectively. And from this scale of numbers and logarithms, the same properties easily follow, as before mentioned.
To compute the logarithm to any of the natural numbers, 1, 2,
3, 4, 5, 6ợc.
Let 6 be the number, whose logarithm is required to be found ; and a the number next less than b, so that bm= 1, the logarithm of a being known ; and let s denote the sum of the two numbers at-b. Then
1. Divide the constant decimal •8685889638, &c. by se and reserve the quotient ; divide the reserved quotient by the square of s, and reserve this quotient ; divide this last quotient also by the
the square of s, and again reserve the quo
, tient ; and thus proceed, continually dividing the last quotient by the square of s, as long as division can be made.
2. Then write these quotients orderly under one another, the first uppermost, and divide them respectively by