Let N be any number and n any positive in teger. Then logo (N 10n)=1ogoN-Fn log,. 10..o. -{-log,. N; and log., (N= 10n) = log,. N - n logo 10 -= N. Now multiplication or division by a power of 10 has only the effect of moving the decimal point, while the logarithm of the product, as just seen, is equal to that of the multiplicant (or dividend) increased (or de creased) by an integer. Accordingly, if two numbers differ only in the position of the decimal point, their logarithms differ only in respect to the integral part (called the char acteristic), the fractional part (called the man tusa) being the same in both. In that fact re sides the chief practical advantage of the Briggs ian system. For example, if logi.2.23 ----- .3483, it follows that log,.22.3 = 1.3483; and log0.00223.... logo (223 χ 10') - 3 + .3483, or 3.3483, as negalive characteristics are often written. It is easy to see that the characteristic of a loga rithm is + n if the number has n + 1 figures before the decimal point, and is -n if the number is a pure decimal in which the point is followed by a-1 zeros. Thus, log023506.054 + a pure decimal, and -4+ a pure decimal.
Logarithmic Series, Calculation of Log arithms. It may be shown by the methods of the differential calculus that if x < 1, loge(1 xl x) =- x - 2 3 5 - x2 4 - - - - . . . . , the logarithmic series. Similarly, loge (1-4 xs xs xs - z 2 3 4 5 . The logarithmic series converges slowly for all but small values of x. It is on that account ill adapted to the com putation of Napicrian logarithms. A series bet ter adapted to such calculation is, however, read ily obtained as follows: From the last two series 1+x it follows that loge =2 + ) 1x at - n 1+x m Put x= so that = m 1-x n ' then log m - 1_ int -n 3 /fa t:\ 1 , n mi-rs m-{-n) 5 km-I-n/ a rapidly converging series that may be used for the calculation of logarithms as follows. For m=2 and n=1, we get log e2 = 0+2 whence loge2=.693147 (correct to six decimal places). For m = 3 and a = 2, the series gives =loge2+21 +RIP+ . . = 1 . 098612. Tak ing m = 5 and n =3, it is found that 1.609438. Then 2 loge2, +loge3, 108e8= 3 and so on. In particular logel 0 +log.5=2.302585. Since logo N= (logeN) (loge10) , it is seen that the common logarithm of any number may be found from the Na pierian longarithm of that number by plying the latter logarithm by 2.302585. This last is called the modulus of common logo rithms. It is obviously possible to calculate logarithms that shall be correct to any pre scribed number of decimal places. Logarithms correct to 3 or 4 places are sufficiently accu rate for all ordinary computations, though tables correct to 5, 6, 7 and even 10 or more places are often employed. By means of any such table can be found the logarithm of any given number and conversely. The number corresponding to a given logarithm is often called the antilogarithm. The advantage of loga rithmic over ordinary computation is easily seen. Thus to find the product of two or more numbers, it suffices to add their logarithms and then to take the antilogarithm of the sum. To
extract any root, say the 7th, of any number, it suffices to divide the logarithm of the number by 7 and to take the antilogarithm of the quo tient. To find the quotient of two numbers, it suffices to subtract the logarithm of the di visor from that of the dividend and to take the antilogarithm of the difference. The cologa rithm of a number is the logarithm of the re ciprocal; thus, colog n =log 1 =logt- log is =0- log n; hence to subtract a logarithm is equivalent to adding the corresponding colog arithm.
Undetermined Coefficients.- Reference was made above to this subject, of which some account will now be given. Let man 1-a,ra-11-aart-2-}- . . . a.--,x+a,,, a rational in tegral function of degree ninx. It can be proved and is here assumed that any such function vanishes for some value of the variable. If Ars) =0, then, by the factor theorem, i(x) =(x-r,)f '(x), where f'(x) . . . If f' (r,)=-0, then f' (x f (x), where f 2 . . . , and hence f (x) (x-r1) (x-rOr (x). By the argument here exemplified it is proved that f(x) may be put in the form f (x).7.-:,704(x-ra) (x - re) . . . ( -ro). Each of the n numbers r,, r,, ,ra causes f (x) to vanish; hence the a numbers are roots of the equation f(x) =0. It can be easily seen that the equa tion f(x) =0 cannot have more than a differ ent roots unless its coefficients are each zero; that is, f(x) cannot vanish for more than different values of .r unless For if f (r.+,)=0, then a.(rn+ r -ri)(rn-Fi (rn+i-rn)=0, but by hypothesis no () = 0, hence a. = 0, and f (x) = atxte-1 ... As the latter is to vanish for more than n - 1 values of x, =0. In like manner it would follow that al =0, ..., an =0. But if the coefficients are each zero, f(x) vanishes for every x, so that if f(x) vanishes for more than n values of x, it vanishes for all x's. Now suppose that a.rs +a,xx--2+ ... +an is to be equal to all values of x, then the function (a.- bo)xn + (a,- bi)xn-1-1- ... +(a. -b.) must vanish for every value of x, and, conse quently, a.= b., ... , a,, b,,. Hence two rational integral functions of degree n in .r are equal for all values of x, i.e., are identical, when and only when the coefficients of like powers of x are equal. This proposition enables us to solve many problems involving the determination of undetermined coefficients. For example, sup pose it required to find the sum of the squares of the first a integers. Assume the identity +(n-1) + -1-qtth, where the coefficientsa, b , ease to be determined. Replacing is by n + 1, we obtain ... (n+ a +b(n+ I) +c(n+ 1)'+ ... By subtracting corre sponding members of the identities, there results the identity an + 1 3dn +d As this relation is to be valid for every value of n, coefficients of lace powers of n must be equal. Hence e=0, f==0, 1=3d, 2=3d+2c, 1=b+c-Fd; hence b=3, c=3, Accordingly, i'+ + + true for every value of is, hence for n = 1, and hence a=0. Therefore P+2' (2n +1).