Consequently the value of x may be expressed by the continued fraction, From the theory of continued fractions it is known that the greater the number of terms taken the more exact the approximation, and that the error may be rendered less than any assigned number. CALCULATION OF THE LOGARITHM OF A NUMBER. Let a=10, b=2; therefore a=b becomes 10=2, and x is the logarithm of 2 in the system whose base is 10. Since log. 10, and log. 10=1, r falls between 0 and 1. Now 238 and 24=16, therefore the value of a is greater than 3 and less than 4. Let a 3+3 ... 10=23 × 2o and = 10 23 × 29 8 23 or 1.25=2o; 1. Next, to satisfy the equality 125=2o, it is necessary that ß<1, let 1 1.25 2 or 1.25"=2. Substituting successively 3 and 4 for y, 1-253 <2 and 1.25">2. Whence the value of y falls between 3 and 4; let y=3+8. ... 1-253 × 1·25*=2, and 1-253 × 1.25* 2 = 1 1 Replacing ẞ in the expression r=- by -= 3+8 or 1.25 1.024. In the same manner in the equality 1·25=1·024, is fractional or of ; whence 1.25=1·024′; &=9+% and 1·0249 × 1·024' 1.25 or 1·0097. Replacing in equation 2 by = Similarly the equality 1·024'=1.0097, by the substitution of for %, gives 1.024=1-0097"; from which are obtained 7=2+0 and 1.0097°=1.0044. The equality 1.0097°=1.0044, by the substitution of for 0, gives 1·0097= 1.0044, λ=2+μ and 1·0044"=1.00087. 1 And the equality 1-0044′′-1-00087, by the substitution of for μ, gives v=5+w; 1.00087"1.000042; whence r=log. 2= =0·3010301, a result of which the six first decimals are correct. 789 2641 222. When the base is 10, 101=10, 102=100, 103=1000; . . . therefore the logarithms of the numbers 10, 100, 1000, general the logarithm of 1000 ... are 1, 2, 3 ...; and in k (that is 1 followed by k zeros, or 10 raised to a power k) is k, or log. 10*=k. To obtain the logarithms of the numbers 2, 3, 5, 7, .... it is necessary to resolve the equations 10=2, 10=3, 10a=5, 10′′=7....; the logarithms of the numbers 4, 6, 8, and in general of composite numbers, are obtained from the logarithms of the prime factors of these numbers by addition (Art. 217). The logarithms of numbers may be computed, with great labour, by the methods of Articles 216, 217, 221; more expeditious processes are derived from the integral calculus. 223. Logarithms being considered as exponents, and calculated for a particular base, it is easy to transform them into logarithms which shall have a different base. Let a denote the base of an existing system of logarithms, a', the base of a new system, r the log. of any number y, in system a', then a'"'=y. Taking the logarithms of both members of this equation in the system a, Log. (a'*')=log. y or x′ log. a'=log. y. 1 I' = a constant quantity. Now is the logarithm of y in the system a' and log. y is the logarithm of y in the system a; therefore whatever be the number y there exists a constant ratio between its two logarithms taken in the two systems. When logarithms are deduced from progressions, this consequence is presented by the progressions themselves. Leth hq: hq hq3 : hq* : . . . . be any geometrical progression; the logarithms of the different terms are, Log. h. log. h+log. q, log. h+2 log. q, log.h+3 log. q, &c. Therefore if numbers are in geometrical progression, their logarithms are in arithmetical progression. Making h=1, the two progressions become, #1 : 9 : q2 : : .... 0. log. q. 2 log. q. 3 log. q. 4 log. q. which correspond with the geometrical and arithmetical series of Article 219. OF TABLES OF LOGARITHMS. 224. The base of the tabular logarithms is 10. Some tables contain the logarithms of numbers from 1 to 10000, others from 1 to 108000 ...., the logarithms being variously given to 5, 6, or 7 decimal figures. The problems to be worked with the tables are two: 1st. To find the logarithm of a given number. 2d. To find the nuinber corresponding to a given logarithm. The resolution of these problems depends on the arrangement of the tables employed, which is far from uniform. The arrangement, however, and manner of using any particular set of tables are generally explained in an introduction, to which it seems best to refer the reader. 225. Applications of logarithms. a. Multiplication. Log. (abc...)=log. a+log. b+log.c+ ... Ex. Let a=497·86, b=76·897, c=67948; required the product abc? Log. 497.86 log. 76.897 log. '67948 =26971072 log. 26013-17=4-4151933 The sum of the tenths is 24-2.4. The sum of the characteristics is 2+1−1=2. Whence, since the 2 carried from the sum of the decimal parts of the logarithms is positive, the characteristic of the log. of the product is 2+2=4; the log. of the product is 4-4151933, and the product nearly 26013-17. b. Division. Log. (a+b)=log. a-log. b. Let a=87946, b=9842. The difference of the logs. of the numbers is 0.9511327, and the number corresponding to this log. is 8-935784, which is the quotient required. The method of subtraction by arithmetical complements is explained, Part I. Art. 42. The process is the same with decimal fractions (and by consequence with logarithms) as with whole numbers. Taking the last example as an instance, Log. 87946=4-9442161 } The arithmetical complement of the log. of 9842=6·0069166 The log. of the result, as before, is 10.9511327 10.0000000 0.9511327 add. subtract. Since log. (a+b)=log. (7)=log. a-log. b, the logarithm of a fraction is obtained by subtracting the log. of the denominator from the log. of the numerator. As an example in which multiplication and division are combined, let it be required to find the product of the fractions 1, 13, and 4? Log. 31 log. 47 =1.4913617 =1.1139434 =1.6720979 log. 75, arith. comp. 8-1249387 8.9208188 8.3187588 29.6419193 Otherwise the logarithms of the numerators may be added into one sum, the logarithms of the denominators into another; and the latter subtracted from the former. c. Formation of powers. Log. (a")=mxlog. a. Let a=25·4, and m=3; then a”=(25·4)3, and m×log. a=3× log. 25·4 ? Log. 25.4 1.4048337 3 multiply. ?... log. (aTM)=8(log. 3—log. 5), Log. 3=0.4771213 log. 5, arith. comp. 9.3010300 But 2-2252344 is the log. of 0167971, which number is, therefore, the 8th power of . When a is a large number, and m>2, tables which give the logarithms of numbers to 6 or 7 figures afford only a rough approximation to the value of a". Gardiner's and Callet's tables contain supplementary tables of the logs. of numbers to 20 figures; with the help of these a near approximation can be obtained. d. Extraction of roots. Log. (a)= n log, a log. 1162049 ? 7 Let a1162049; then log. (√1162049)= Log. 1162049=6·0652254. of this log. is 0.8664607. The number corresponding to this quotient is 7.352932; therefore 1162049=7.352932. 2d Ex. Required the value of the expression x=W Log.x=log. (W)=(log. 13-log. 27). Log. 13 log. 27, arith. comp. 8.5686362 1.6825796 add. To render the negative characteristic of this log. divisible by 11 without altering the value of the expression; since -10+10=0; if -10 is added to the characteristic, and +10 to the mantissa, Let a=0963, b=24958, c='008967. Log. a(='0963), arith. comp. 11.0163737 Hence the log. of a fourth proportional to three given numbers is obtained by adding together the logs. of the second and third terms, and subtracting the log. of the first term from the sum. 2d Ex. Let x= (a2—b2)3a √(a+b)√cd Log. r=log. ? (a2-b2)3a-log./(a+b)√cd. But log. (a2-b2)3a= log. [(a2—b2)3a]=}[log. (a+b)+log. (a−b) +log. 3+log. a], and log. (a+b)√/cd=3[log. (a+b)+ log.c+log.d]; ..log.x=log. (a+b)+log. (a−b)+log. 3+log.a]-[log. (a+b)+ log.c+log.d]. Let a=60, b=15, c=16, d=9. The numerical value of x is log.x=log. (60+15)+log. (60-15)+log. 3+log. 60]— or log.x=log 75+log. 45+log.3+log. 60]-[log. 75+ log. 16+ log. 9]. Log. 75 1.8750613 log. 45=16532125 log. 30.4771213 3)5-7835464 Log. 75=1.8750613 log. 16=0.6020600 log. 9=0.4771212 log. 60=1.7781513 1.9278488 Whence log. x=1.9278488-1-4771212=0.4507276, and x 2.823108, the value required. f. Exponential equations. The equation a=b, in which the unknown quantity is an exponent, is termed an Exponential equation. By the resolution of the exponential equation a=b, the logarithm of the number b, in the system whose base is a, is obtained (Art. 221). Conversely, tables of logarithms afford the means of resolving any exponential equation in which a, b represent numbers: For since ab log. (a*)=log. b, or x log. a=log.b, .*.x- log. a' a=b is an exponential equation of the first order. abc, abo=d, . ... are exponential equations of the 2d, 3d, . . . order. The expression a signifies that b is raised to the power x, and that a is then raised to the power br. Since ac, log. (ab)=log. c; or be log. a=log. c. Therefore log. (b log. a)=log. (log. c); or log. (b)+log. log. a=log. log. c. Now log. (b)=x log. b; and since log. a, log. c are decimal fractions, their logarithms can be taken like the logarithms of any other numbers. Therefore a log. b+log. log. a=log. log c; and x= log. log. c-log. log. a If the value of an algebraic expression is negative, the calculation cannot be made by logarithms without changing the signs of the terms of the quan |