Let m be the larger, and n the smaller of the two numbers. Then m= 10" x n, where r is some integer. :. logio m= log10 10” + logio n (Law I.)=r + log10 n. Now, since the mantissæ of all logarithms are written positive, the addition of the integer r will only affect the characteristic. Hence the mantissa of logio m = the mantissa of logio n. Examples. The logarithm of 1075-06 is 3.0314327 (the characteristic being seen by the rule of the characteristic). Now 107506=1075.06 x 100, .: log 107506=log 1075.06+2=5•0314327; and 1•07506=1075-06: 1000, .: log 1•07506=log 1075-06 - 3=·0314327; and ·00107506=1075.06 : 1000000, .. log .00107506=log 1075.06 – 6=3.0314327. The student should observe that each of the above characteristics follows the characteristic rule. 263. The two rules above proved make it unnecessary to publish in the tables either (1) The characteristic of the logarithm of any given number, for this can be seen by inspection of the given number: or (2) The position of the decimal point in the given number, for this does not affect the mantissa of the logarithm which is alone tabulated. Hence what we find in the table is e.g. No=59543. Log = 7748307. This means that log .0059543 = 3.7748307, &c. &c. Hence a single reference in the table gives us the logarithms of an indefinite number of numbers. To § 4. THE USE OF MATHEMATICAL TABLES. 264. To explain the mode of operating upon numbers whose integral part is negative and decimal part positive. An example in each of the operations of addition, subtraction, multiplication, and division will suffice. 7.432196 7.432196. x 5 7.432196 Addition. Add the decimal parts in the ordinary way. When we reach the integral parts which are negative, we have +1-7-5=-11. Subtraction. Subtract the decimal parts in the ordinary way. When we reach the integral parts which are negative, we have -11-(-5)-1=-7. Multiplication. Multiply the decimal part in the ordinary way. When we reach the integral part which is negative, we have (-7 ~ 5) + 2 =-35+ 2 =- 33. Division. Find the multiple of the divisor next higher (instead of next lower) than the integral part of the dividend. Thus -33-5=(-35+2) - 5 = -7 +(2-5). Then proceed as in ordinary division. 265. To explain the method, when the logarithms of certain numbers are given, of finding the logarithms of other numbers connected with those given. [When the base is not indicated, it is understood here that the base is 10.] The logarithms of different numbers to the same base are connected by the laws given in Art. 246. It is always necessary to resolve a given number into its factors in order to find what numbers are logarithmically connected with the given number. Thus: 6 = 3 x 2; 150 = 3 x 2 x 5 x 5 = 3.2.52; 1260 = 22. 32.5.7. The student should at once accustom himself to resolve any number in this way into its prime factors. Suppose then that we have given log 2 •3010300 and log 3 = •4771213, we can then find the logarithm of any number of the form 2m. 3". For example : Log 6=log (3 x 2)=log 3+log 2 (by Law I.) =:3010300+ •4771213=7781513, =3 log 3+2 log 2 (by Law III.) =1.4313639+ :6020600=2.0334239, log : =log 2-1= -log 2 (by Law III.) - •3010300= -1+6989700=1.6989700, 9 32 = log = log 32 - log 25 (by Law II.) 32 26 = . =•9542426 – 1.5051500=1.4490926, =} (log 3+log 2)+} (log 3+2 log 2) log 266. The factors of 10—the base itself of our logarithmsare 2 and 5. Thus, when either log 2 or log 5 is given, we may find the other : for log 5 = log (10-2) = log 10 - log 2 = 1 - log 2. log 2 = log (10 = 5) = log 10 – log 5 =1 – log 5. It is important to remember this. Example. Given log 2= 3010300, find log 50. log 50=log 5 +log 10=log 10 – log 2 +log 10 =2 log 10 – •3010300=1.6989700. The Principles of Approximate Calculations. 267. When a number is given to (say) 7 decimal places, it is generally understood that the value given is nearer to the true value than any other number containing the same number of decimal places. Thus, when we say log 3 = .4771213, what is meant is that log 3 is nearer to ·4771213 than to •4771212 or •4771214; in other words it lies between ·47712125 and .47712135. Hence, the error involved in taking a number calculated to 7 decimal places is less than :00000005, i.e. less than 1 of 10-7. When we calculate a number by adding or subtracting such approximate numbers together, we add together the possible errors. Hence the actual error in the resulting sum may be greater than in any of the numbers added or subtracted. For example log 3 is really a very little greater than ·47712125; hence (on examining a table of logarithms) we shall find that log 9, i.e. 2 log 3, is (not :9542426 but) .9542425; that log 81, i.e. 4 log 3, is 1.9084850; that log 729, i.e. 6 log 3, is 2.8627275; and so on. Thus the error involved in calculating log 729 from log 3 is increased 6 fold and is about 3 x 10-7 instead of 1 of 10-7. (See Art. 34.) 268. It is sometimes convenient to drop the last figure in an approximately calculated number. In such a case, the principle explained in the last article shows that the last figure retained should be increased by unity, if the figure dropped is greater than 5, or if the figure dropped is equal to 5 and the true value exceeds the given value. Thus, dropping the last figure of 4:321976, we have 4:32198 ; for this is clearly nearer to the true value than 4.32197. Again, dropping the last figure of 4:321975, we should write 4:32198 or 4.32197 according as the approximation 4:321975 is less or greater than the true value. 269. The student is advised to make use of some book of Mathematical Tables : such as Chambers's. The following lists of corresponding numbers will be found : (1) The mantisce of logarithms, calculated to 7 places of decimals, which correspond to the 100,000 numbers between 0 and 100,000, successively increasing by unity. (2) The trigonometrical ratios, calculated to 7 places of decimals, which correspond to the 5400 angles between 0° and 90°, successively increasing by 1 minute. (3) The logarithms (increased by 10), calculated to 7 places of decimals, which correspond to the ratios of these angles. 270. If it is required to find a number corresponding to any number given in the tables, mere reference to these tables is sufficient. But if it is required to find a number corresponding to some number intermediate between two consecutive numbers in the table, we must 'interpolate' according to the following rule; the foundation and limits of which will be explained in a later chapter. Rule of Proportional Differences. 271. The small differences between the values of one varying quantity are approximately proportional to the small differences between the corresponding values of another quantity which varies with the first. Thus, if L and N are the values corresponding to successive values l and n given in the table, and if we have to find the value M corresponding to m, which lies between 1 and n, we have the equation M-L N-L Thus the difference M – L, which has to be added to the given value L to find M, (N – L). 2 m-/ n |