Hence, to find the common logarithm of any number, multiply the Naperian logarithm of the number by the modulus of the common system. It now remains to find the Modulus of the common system. Hence, the modulus of the common system is equal to the common logarithm of any number divided by the Naperian logarithm of the same number. But the common logarithm of 10 is 1, and we have calculated the Naperian logarithm of 10, (Art. 375); therefore, which is the modulus of the common system. Hence, if N is any number, we have com. log. N=.4342944× Nap. log. N. On account of the importance of the number A, its value has been calculated with great exactness. It is A=.43429448190325182765. ART. 377. To calculate the common logarithms of numbers directly. Having found the modulus of the common system, if we multiply both members of equation (7), Art. 374, by A, and recollect that AX Nap. log. N= com. log. N, the series becomes 1 1 log. (x+1)= log. x+2A {22+1+3(2x+1)s+5(2x+1)* Or, by changing z into P, for the sake of distinction, and putting B, C, D, &c., to represent the terms immediately preceding those in which they are used, we have B 3C 2P+13(2P+1)2+5(2P+1)2 2A log. (P+1)= log. P+, 5D TE 9F +7(2+1)2+9(2+1)2+11(2P+1)2+, &c. We shall now exemplify its use in finding the logarithm of 2. Here P=1, and 2P+1=3. Exercise. In a similar manner let the pupil calculate the common logarithms of 3, 5, 7, and 11. For the results to 6 places of decimals, see the Table, page 316. ART. 378. To find the base of the Naperian system of loga rithms. If we designate the base by e, we have, (Art. 376), log. e : log', e : : A : A'. But A=.4342944, A'=1, and log'. e=1, (Art. 367); hence, log. e:1::.4342944 :1, But since we have explained the method of calculating common logarithms, they are supposed to be known, and we may use them to obtain the number of which the logarithm is .4342944, which we shall find to be e=2.71828128. We thus see that in both the common and the Naperian systems of logarithms, the base is greater than unity. Brigg's logarithms are used in the ordinary operations of multiplication, division, &c., and hence are called common logarithms. Napier's logarithms are used in the applications of the Calculus. These are the only systems much used. ART. 379. The student may prove the following theorems : 1. No system of logarithms can have a negative base, or have unity for its base. 2. The logarithms of the same numbers in two different systems have the same ratio to each other. 3. The difference of the logarithms of two consecutive numbers is less, as the numbers themselves are greater. SINGLE AND DOUBLE POSITION. NOTE. On account of the use made of Double Position in the solution of exponential and other equations, it becomes necessary to explain the principles on which it is founded. We shall also explain Single Position. ART. 380. SINGLE POSITION.-The Rule of Single Position is applied to the solution of those questions in which there is a result which is increased or diminished in the same ratio with some unknown quantity which it is required to find. Of this class are all questions which give rise to an equation of the form ax=m (1). If we assume x' to be the value of x, and denote by m' the result of the substitution of x' for x, we have ax'=m' (2). Comparing equations (1) and (2), we have m':m:: ax': ax: x': x; that is, As the result of the supposition is to the result in the question, so is the supposed number to the number required. ART. 381. DOUBLE POSITION.- The Rule of Double Position is applied to those questions in which the result, although it is dependent on the unknown quantity, does not increase or diminish in the same ratio with it. The class of questions to which it is particularly applicable, gives rise to an equation of the form ax+b=m (1). If we suppose x' and x" to be near values of x, and e' and e" to be the errors, or the differences between the true result and the results obtained by substituting x and x" for x, we have If we subtract equation (1) from (2), and (3) from (2), we have By subtracting equation (1) from (3) we also find a(x"—x)=e", and thence, x- x". -X e' -e" e" (7). Hence (Art. 263), The difference of the errors is to the difference of the two assumed numbers, as the error of either result is to the difference between the true result and the corresponding assumed number. When the question gives rise to an equation of the form ax+b=m, this rule gives a result absolutely correct; but when the equation is of a less simple form, as in exponential equations (Art. 383), the result obtained is only approximately true. Cor. The value of x, found either from equation (6) or (7), is This, expressed in ordinary e'x'-e'"'x' language, furnishes the common arithmetical rule. EXPONENTIAL EQUATIONS. ART. 382. An exponential equation is an equation in which the unknown quantity appears in the form of an exponent or index, as ax=b, xx=a, al*=c, &c. Such equations are most easily solved by means of logarithms. Thus, in the equation ax-b if we take the logarithms of both members, we have x log. a= log. b, Ex. 1. What is the value of x in the equation 2*=64? Here x log, 2= log. 64. ART. 383. If the equation is of the form xa, the value of x may be found by Double Position as follows: Find by trial two numbers nearly equal to the value of x ; sub stitute them for x in the given equation, and note the results. Then, As the difference of the errors; Is to the difference of the two assumed numbers ; So is the error of either result; To the correction to be applied to the corresponding assumed num ber. Ex. 1. Given x=100, to find the value of x. The value of x is evidently between 3 and 4, since 33-27, and 4' 256; hence, taking the logarithms of both sides of the equation, we have x log. x= log. 100=2. By trial, we readily find that x is greater than 3.5, and less than 3.6; then let us assume 3.5 and 3.6 for the two numbers. First Supposition. Second Supposition. x=3.5; log. x=,544068 | x=3.6; log. x=.556303 x. log. x true no. error multiply by 3.6 we find =1.904238 x. log. x =2.002690 =2.000000 true no. =2.000000 Diff. results: Diff. assumed nos. :: Error 2nd result: Its cor. .098452 : Hence, x=3.6-.00273-3.59727 nearly. : .00273 By trial we find that 3.5972 is less, and 3.5973 greater than the true value; and by repeating the operation with these numbers we would find x=3.5972849 nearly. 6. How many places of figures will there be in the number expressing the 64th power of 2? 7. Given ab+d=c, to find x. Ans. 20. |