THEOREM III. The logarithm of the quotient of two numbers is the difference between the logarithms of the numbers. THEOREM IV. The logarithm of the real nth root of a number is the logarithm of the number divided by n. and EXERCISES Given log10 2 =. .301, log10 5.699, log10 7 = .8451, find 1. log (73. √5).* Solution : By Theorem I, log (√73. √5) = log √73 + log √5. By Theorems III and IV, Now = log 7+log 5. = (.8451) = .50706, Adding, log 7 + 1⁄2 log 5 = log (√78. √5) = .85556 * Where no base is written it is assumed that the base 10 is employed. 210. Common system of logarithms. For purposes of computation 10 is taken as a base, and unless some other base is indicated we shall assume that such is the case for the rest of this chapter. We may write as follows the equations which show the numbers of which integers are the logarithms. Assuming that as x becomes greater log x also becomes greater, we see that a number, for example, between 10 and 100 has a logarithm between 1 and 2. In fact the logarithm of any number not an exact power of 10 consists of a whole-number part and a decimal part. Thus since Since 1033421 <104, log 3421 3. + a decimal. 10-3<.0023<10-2, log .0023= 3. + a decimal. The whole-number part of the logarithm of a number is called the characteristic of the logarithm. The decimal part of the logarithm of a number is called the mantissa of the logarithm. The characteristic of the logarithm of any number may be seen from the above table, from which the following rules are immediately deduced. The characteristic of the logarithm of a number greater than unity is one less than the number of digits to the left of its decimal point. Thus the characteristic of the logarithm of 471 is 2, since 471 is between 100 and 1000; of 27.93 is 1, since this number is between 10 and 100; of 8964.2 is 3, since this number is between 1000 and 10,000. The characteristic of the logarithm of a number less than 1 is one greater negatively than the number of zeros preceding the first significant figure. Thus the characteristic of the logarithm of .04 is 2; of .006791 is — 3; of .4791 is 1. It must constantly be kept in mind that the logarithm of a number less than 1 consists of a negative integer as a characteristic plus a positive mantissa. To avoid complication it is desirable always to add 10 to and subtract 10 from a logarithm when the characteristic is negative. Thus, for instance, instead of writing the logarithm 3.4672 we write 103 + .4672 — 10, or 7.4672 10. This is convenient when for example we wish to divide a logarithm by 2, as by Theorem IV, § 209, we shall wish to do when we extract a square root. Since in the logarithm 3.4672 the mantissa is positive, it would not be correct to divide — 3.4672 by 2, as we should confuse the positive and negative parts. This confusion is avoided if we use the form 7.467210, and the result of division by 2 is 3.7336-5, or 8.7336 10. The actual logarithm which is the result of this division is - 2 + .7336. THEOREM. Numbers with the same significant figures which differ only in the position of their decimal points have the same mantissa. or If we multiply both numbers of this equation by 100, we have 10210x 10x+2 2431, = = x + 2 = log 2431. Thus the logarithm of one number differs from that of the other merely in the characteristic. In general numbers with the same significant figures are identical except for multiples of 10. Hence their logarithms differ only by integers, leaving their mantissas the same. Thus if log 47120.4.6732, log 47.12 = 1.6732, and log.004712 7.6732-10. == 3.6732, or EXERCISES If log 2 = .3010, log 3.4771, log 7 = .8451, find 211. Use of tables. A table of logarithms contains the mantissas of the logarithms of all numbers of a certain number of significant figures. The table found later in this chapter gives immediately the mantissas for all numbers of three significant figures. In the next section a method is given for finding the mantissa for a number of four figures. Hence the table is called a four-place table. Before every mantissa in the table a decimal point is assumed to stand, but in order to save space it is not written. To find the logarithm of a number of three or fewer significant figures we apply the following RULE. Determine the characteristic by rules in § 210. Find in column N the first two significant figures of the number. The mantissa required is in the row with these figures. Find at the top of the page the last figure of the number. The mantissa required is in the column with this figure. When the first significant figure is 1 we may find the logarithm of any number of four figures by this rule from the table on pp. 248, 249 if we find the first three instead of the first two figures in column N. = = 212. Interpolation. We find by the preceding rule that log 2440 3.3874, while log 2450 3.3892. If we seek the logarithm of a number between 2440 and 2450, say that of 2445, it would clearly be between 3.3874 and 3.3892. Since 2445 is just halfway between 2440 and 2450, we assume that its logarithm is halfway between the two logarithms. To find log 2445, then, we look up log 2440 and log 2450, take half (or .5) their difference, and add this to the log 2440. This gives If we had to find log 2442 we should take not half the difference but two tenths of the difference between the logarithms of 2440 and 2450, since 2442 is not halfway between them but two tenths of the way. This method is perfectly general, and we may always find the logarithm of a number of more than three figures by the following RULE. Annex to the proper characteristic the mantissa of the first three significant figures. Multiply the difference between this mantissa and the next larger mantissa in the table (called the tabular difference and denoted by D) by the remaining figures of the number preceded by a decimal point. Add this product to the extreme right of the logarithm of the first three figures, rejecting all decimal places beyond the fourth. In this process of interpolation we have assumed and used the principle that the increase of the logarithm is proportional to the increase of the number. This principle is not strictly true, though for numbers whose first significant figure is greater than 1 the error is so small as not to appear in the fourth place of the logarithm. For numbers whose first significant figure is less than 2 this error would often appear if we found the fourth place by interpolation. For this reason the table on pp. 248, 249 gives the logarithms of all such numbers exact to four figures, and in this part of the table we do not need to interpolate at all. |