ef. 2.-The value of a power is the arithmetical result, as indicated by that power. Cor. Hence the value of a power does not exhibit any trace of that power. Def. 3.-The exponent of the power of a number is called the logarithm of the value of that power. Def. 4.-A system of logarithms is the numbers arising in the exponent of a power according to every value that may be given to that power, while the base is any constant number greater than unity. Def. 5. -The base of a system of logarithms is the same as the base of the power. Illustration.--Thus, let y=a*, and if y be made successively equal to 1, 10, 100, 1000, 10000, &c. And the base a equal to 10, we shall have y = a successively, 1 = 10°, 10 = 101, 100 ≈ 102, 1000 = 103, 10000 104, and so on. Then 0, 1, 2, 3, 4, are logarithms of 1, 10, 100, 1000, 10000, and bclong to a system of logarithms whose base is 10; or, if y be made suc1 1 1 and so on, the base being still equal to 10, 10' 100' 1000' 10000' cessively 1 =10-3, 1 10000 1 1 1 1 1 1 10-4, &c. that is = 10 101 100 102' 10000 = = &c. so that the logarithms of numbers less than unity 103' 10000 104' are negative. Def. 6.-When a system of logarithms has the number 2.71828, &c. for its base, that system is called the Naperian System,* and every logarithm taken from the Naperian system is called the Naperian logarithm of its corresponding number Def. 7.-When a system of logarithms has the number 10 for its base, that system is called the Briggian System, † and every logarithm belonging to the Briggian system is called the Briggian logarithm of its corresponding number. Notation. The number 2.71828, &c. is represented by e. The logarithm of any number indicated by an italic letter is represented by the corresponding capital letter. Also, the logarithm of any number, a or y, is denoted by log, x or log. y. Theorem 1.-The sum of the logarithms of any two numbers, x and z, is equal to the logarithm of their product. For, since in the equations x = a*, and z = a2, X is the logarithm of x, and Z is the logarithm of z, by Definition 1: multiply the corresponding sides of these two equations together, and we shall have xz = axa2 = aX+Z; but, by Definition 3, the exponent * From Napier, the inventor of logarithms. + From Henry Briggs, who was the first that changed Napier's system, by introducing 10 for the base instead of 2.71828, &c. X+Z is the logarithm of az. Therefore, the sum of the logarithms of any two numbers is equal to the logarithms of their product. Q.E.D. Cor. Hence it is evident that the sum of the logarithms of any number of numbers is the logarithm of the product of these numbers. Theorem 2.-The difference of the logarithms of any two numbers, a and z, is equal to the logarithm of their quotient. For, since xa and z = a2; therefore = ax qz = ax-z; but 2 az by Definition 3, the exponent X Z of the power of which a is the base, is the logarithm of: whence the difference of the logarithms of any two numbers is equal to the logarithm of their quotient. Q.E.D. Theorem 3.-The logarithm of the nth power of any number x is equal to n times the logarithm of that number. Because x = ax, take the nth power of both sides of the equa. tion, and we shall have x" = a′′X; but nX is the logarithm of x"; therefore the logarithm of the nth power of any number is equal to n times the logarithm of that number. Q.E.D. Theorem 4.-The logarithm of the nth root of any number x is equal to the nth part of the logarithm of that number. Because xa, take the nth root of both sides of this equa X tion, and we shall have xa*: but X n I is the logarithm of ai therefore the logarithm of the nth root of any number is equal to the nth part of the logarithm of that number. Q.E.D. Theorem 5.-The logarithm of any number, 1 + x, will be equal 303 3 X4 4 5 &c.) For the expansion of : (1+x) or of : (1 + 2) so as to ad mit of the property : (1+x) + 4 : (1 + ≈) = ? : {(1+x) × (1 + z) } is the series announced by this theorem; but (by Theorem 1, present article of Logarithm,) log. (1 + x) + log. (1 + 2) = log. {(1 + a) × (1+2)}; and, as this is included in the functional equation, we shall then have the log. (1 + x) = M Cor.-Hence, if x is equal to zero, the series expressing the value of the logarithm of (1 + z) will vanish, and the logarithm of 1 will remain: hence the logarithm of 1 is zero, or nought. LOGARITHMS. Theorem 6.-The logarithm of the number 1 + x is equal to the For, if in the serical equation, log. (1 + x)= M 2x 2 then log. (1-x)= &c.); subtract the latter equa we shall have the equation log. (1 + x) + 3 7 Theorem 2, log. (1 + x) — log. (1—x) = log. 1 + x log. 1 x + &c.); but, by =M (2x+ +255 + + &c.) Theorem 7.-The logarithm of the number z + v is equal to the sum of the series, log. z + M For, if, in the equation, log.¦±= M (x + x3 + + 3 5 + 'z + v) — log. z = M { + &c.} ; ( 1(2z + v)3 3(2z+v)5 5(2z + v)s ; or, by transposing log. z, we shall have log. (≈ + v) = Cor.-If v = 1 and n➡z + 1, then the log. (≈ + 1) = log. n and log. SCHOLIUM.-The series announced in Theorem 5 is of very little use in the calculation of logarithms, owing to the very slow degree of the convergency of its terms. But, in large numbers, the series announced in the last Theorem, or its Corollary, must be used; for, having the logarithm of a low number, that of a higher one may be found by the series PROBLEM. To find the logarithm of a low number by the equation log. 1 + x x3 x7 = M (x + + + + &c.) according 7 to the Briggian system. 1 + x Put equal to the proposed number, which let it be n, then in the series, and, instead of M, substitute 2 x .43429448 = .86858196, and we shall have the equation Reduce the vulgar fractions to decimals, add the decimals of the terms together, and the sum will be the logarithm of the number required; or, according to the following Rule.-Multiply the uumber 86858896 by the numerator of the value of x, divide the product by the denominator of the same value. Multiply the quotient by the square of the numerator, and divide the product by the square of the denominator. Continue this last operation of multiplying and dividing the last quotient by the squares of the terms of the fraction, keeping the last or right-hand figure under the last of that above, until the last quotient be very small. Divide the first quotient by 1, the second by 3, the third by 5, &c. until the last divisor cannot be contained in the last of the preceding quotients, keeping the right-hand figure of any quotient under that above, as before; then the sum of the last quotient will be the lo garithm of the number required. Example 1.-Find the logarithm of the number 2. Heren 2, therefore n1 = 2 11, and n + 1 = 2 + 1 = 3; whence (n − 1)2 1, and (n + 1)2 = 9. The operation will then be as follows: PROBLEM. To find the logarithm of a number, the logarithm of the next less number being given.-In the equation log. n = log. (n. 1) M { 2 2 + 1(2n-1) 3(2n- 13 Then proceed according to the following } Rule.-For n substitute the number, the logarithm of which is required. Divide the number .863589, first, by the value of 2n the quotient by (2n-1), and so on. . then. Proceed in the same manner, always dividing the last quotient by (2n-1)2. Divide the quotients, as they succeed each other, by the corre sponding numbers of the series of odd numbers, 1, 3, 5, &c. To the sum of the quotients add the logarithm of the next lower number, and the total sum wili be the logarithm of the number required. Example. Find the logarithm of 19, the logarithm of 18 being 1.255273. Here n = 19; whence 2n- 1381 = 37, and 372 = 1369, whence the operation .868589 ÷ 37 = .023475 17 Again, .023175 EXPLANATION AND USE OF THE PROBLEM 1.-To find the logarithm of any number by the table. 1. When the given number is less than 100.-Look for the given num. ber under N in the first column of the first page of the table, and directly opposite to it is its logarithm: thus the log of 98 is 1.991226. 2. When the given number is between 100 and 1000.-Find in some of the following pages, the given number in the first column under N, and opposite to it in the next column, marked 0 at the top, is the decimal part of the logarithm required, before which put an index less than the number of figures; thus log. of 448 is 2.651278. 3. When the given number consists of 4 places.-Find, as before, the first three figures of the given number in some of the columns on the left hand, and the fourth figure at the top or bottom of the page; then directly under the fourth figure, and in a straight line with the three first figures on the left, will be the decimal part of the logarithm sought, before which put the index 1 less than the figures; thus, the log, of 5704 is 3.760724. |