38. A, B, and C are to share 1000001. in the proportion of 1, 1, and 1, 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|f£, and B's 42857 47 LOGARITHMS. LOGARITHMS are numbers so contrived, and adapted to other numbers, that the sums and differences of the former shall correspond to, and show, 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. 0, 1, 2, 3, 4, 5, 6, indices, or logarithms. Thus 1, 2, 4, 8, 16, 32, 64, geometric progression. Or Or 6, indices, or logar. {1, 3, 9, 27, 81, 243, 729, geometric progress. [0, 1, 2, 3, 1, 10, 100, 1000, 10000, 100000, geom. prog. Where it is evident, that the same indices serve equally for any geometric series; and consequently there may be an endless variety of systems pf 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 proportional logarithms, 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 S; and the numbers 4 and 8, or the terms corresponding to those indices, being multiplied together, make 33, which is the number answering to the index 5. In like manner, if any one index be subtracted from another, 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 4=2; and the terms corresponding to those indices are 64 and 16, whose quotient =4; which is the number answering 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 =6; which is the logarithm of 64, or the third power of 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 of 64. The logarithms most convenient for practice are such, as are adapted to a geometric series, increasing in a tenfold proportion, as in the last of the above forms; and are those, which are to be found at present in most of the common ta bles of logarithms. The distinguishing mark of this system 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 1 is —1; that of '01 is 2; that of '001 is 3, &c. the logarithm of 1 being 0 in every system. Whence it follows, that the logarithm of any number between 1 and 10 must be 0 and some fractional parts; and that of a number between 10 and 100, 1 and some fractional parts; and so on, for any other number whatever. And since the integral part of a logarithm, thus readily found, shows the highest place of the corresponding number, it is called the index, or characteristic, and is commonly omitVOL. Is Dd ted 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",then n is log. of N; where n may be either positive or negative, or nothing, and the root r any number whatever, according to the different systems of logarithms. When n is 0, then i is =1, whatever the value of r is; which shows, that the logarithm of 1 is always 0 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 1 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 Nor 2'718, &c.|". 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 n 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 al«o, if 50 be=10169897, then is l'69897 the common logarithm of 50. And in general the following decuple series of terms, viz. 10, 10, 10, 101, 10, 10~\ 102, 10~3. ,10-4 or 10000, 1000, 100, 10, 1, *1, '01, '001, '0001, have 4, 3, 2, 1, 0, −1, −2, —3, —4, for their logarithms, respectively. And from this scale of numbers and logarithms, the same properties easily follow, as beforementioned. PROBLEM. To compute the logarithm to any of the natural numbers, 1, 2, 3, 4, 5, &c. RULE. Let b be the number, whose logarithm is required to be found; and a the number next less than b, so that ba=1, the logarithm of a being known; and let s denote the sum of the two numbers a+b. Then 1. Divide the constant decimal 8685889638, &c. by s, and reserve the quotient; divide the reserved quotient by the square of s, and reserve this quotient; divide this last quotient also by the square of *, and again reserve the quotient; 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 the odd numbers, 1, 3, 5, 7, 9, &c. as long as division can be made; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, and so on. 3. Add all these last quotients together, and the sum will be the logarithm of ba; therefore to this logarithm add also the given logarithm of the said next less number a, so will the last sum be the logarithm of the number b proposed. That is, log. of b is log. a + "x : 1+ - + - + - + &c. S 1 1 1 382 584 756 where n denotes the constant given decimal 8685889638, &c. EXAMPLES, Example 1. Let it be required to find the logarithm of the number 2. Here the given number b is 2, and the next less number a is 1, whose logarithm is O; also the sum 2+1=3=«, and its square s9. Then the operation will be as follows. |