404. We have already seen (Art. 394.) that log. c2= We have also 2 log. c, and generally that log. cn log. c. seen reciprocally that log./c= log. c, and generally that log. /c= log.c n From hence is derived the following general rule; viz. that the logarithm of the nth power of any number is found by multiplying its logarithm by n, and that the logarithm of the nth root of any number is found by dividing its logarithm by n. If, therefore, it were required to find the 10th power of 2, we shall have log. 210=10x log. 2=3.0103000= log. 1024 (). Therefore 210=1024. 405. Again, if it were required to find the square root of 10, 10 1.0000000 we shall have log. 10*= log. 2 2 =0.5000000, = log. 3.16228, the square of which is only greater than 10 406. We have seen (Art. 398.) that the logarithms of all numbers in decuple proportion, preserve the same decimal number, and vary only by their characteristic. From a consideration of this property, which belongs to the system of logarithms, which is founded upon the root a being assumed 10, we are enabled to determine the characteristic of the logarithm from the value of the number, and reciprocally to find the value of the number from the characteristic of the logarithm. For this reason, in the tables of common logarithms, the digits of the number and the decimal logarithm (*) For tables of the logarithms of all numbers from 2 to 108000, and for the method of finding the logarithms of prime numbers, see Dr. Hutton's Mathematical Tables. Ed. Lond. 1822. need need only be registered. For instance, the decimal logarithm of 23000 is .3617278. Now since 23000 lies between 10000 and 100000, we know that the characteristic is 4, and therefore that its logarithm is 4.3617278. By placing a decimal point before the last figure, or dividing it by 10, it will become 2300.0, which lies between 1000 and 10.000, and its logarithm is therefore 3.3617278. Now if the logarithm 1.3617278 were given to find the corresponding number, since we know that the decimal figures belong to the digits 23000, and from the characteristic i we know that the number must lie between 10 and 100, the figures 23000 must be so pointed that the number required shall lie between 10 and 100, which will therefore become 23.000 or 23. 407. Now, if we had the series a, a, a3, a1z, &c. .a", the logarithms of this series would be x, 2x, 3x, &c. .... Nx. From hence we infer, that if a series be in geometrical progression, their logarithms will be in arithmetical progression. 4x, 408. When the characteristic of a logarithm is negative, it is expressed by the sign being placed over it, to distinguish it from the decimal part. Thus the logarithm of 50 being 1.69897, we shall have log. 5=0.69897, and log. 5 =1.69897; log. ·05=2.69897. This expression may be either reduced altogether to a negative form, or it may be made positive, by the following methods: In the first case, making the characteristic less by 1, and beginning with the left hand, subtract each figure of the decimal from 9, except the last, which must be subtracted from 10. Thus the expression 2.69897 may be reduced to -1.30103, which is wholly negative. In the second case, join to the tabular decimal the complement of the index to 10, i. e. increase the indices in the scale by 10; thus log. 2.69897 would be expressed by 8.69897. CHAP. XLIX. ON THE APPLICATION OF LOGARITHMS TO ARITHMETICAL CALCULATIONS. 409. LOGARITHMS are of no particular use in the ordinary operations of arithmetic, but in the involution and evolution of powers, and in their application to complicated calculations, they will be found to be of infinite service, from the peculiar properties which, in the preceding chapter, they have been shown to possess. For instance, in the Rule of Three, if the terms be large, instead of multiplying the second and third terms together, and dividing by the first, we have only to add together the logarithms of the 2d and 3d terms, and to subtract from their sum the logarithm of the 1st, in order to find the 4th. This process is again rendered more easy, if, instead of subtracting any logarithm, we add its complement; i.e. the logarithm of the reciprocal of the given number; the readiest method of finding which, is to begin with the left hand and to subtract every figure from 9, except the last significant figure on the right hand, which must be subtracted from 10. At the conclusion of the calculation, however, it must be remembered that, for every complement that has been so added, the indices or characteristics have been increased by 10, and that, consequently, from the last sum of the indices, 10 must be subtracted for every complement that has been added. EXAMPLE 1. Given, 9: 37:: 4o to find a fourth proportional. Assuming for the number required, we have, x=7 log. 3+6 log. 4-4 log. 9, or + comp. 4 log. 9. Now 7 log. 33.3398491 6 log. 43.6123600 Comp. 4 log. 9=6.1830300 13.1352391 Subtracting 10=3.1352391= log. 1365.333, &c. .. 1365 is the proportional required. EXAMPLE 2. Find a third proportional to /2 and ✔✅10. Let x be the number required. =1-0.06020600=0.9397940 log, 8.70556, which is the third proportional required. EXAMPLE 3. 342 x 559 × 63 Find the value of the fraction, 781 x 432 Let a be the value required, then (Art. 393.) x=log. of the numerator-log. of the denominator. Now log. numerator = log. 342+ log. 559+ log. 63 log. 3422.5340261 log. 559 2.7474118 log. 631.7993405 =7.0807784 log. denominator log. 781+log. 432 log. 7812.8926510 log. 4322.6354837 5.5281347 Hence 7.0807784-5.5281347=1.5526437= log. 35.69891, which is therefore the value of the fraction required. |