10-2-.01 10-3.001 10-.0001 etc. Note that the numbers on the right are positive, decreasing toward zero as the exponents decrease. This indicates, first, that no real power of ten is a negative number, which means further that negative numbers have no real logarithms. The table indicates, secondly, what might be expected for the logarithms of numbers between zero and one. c. For example, the number .06 is between .1 and .01, and one expects the power of 10 giving .06 to be between (-1) and (−2). This logarithm could be given either as 1 minus a decimal fraction or as 2 plus a decimal fraction. The latter method is preferred and must always be used because the mantissa, as determined by the sequence of digits, in this case is the one which appears in the tables. d. The numbers .3, .45, and .1893 are all between .1 and 1; and thus by the preceding table of negative powers of 10, the characteristic of their logarithms is (-1). From the information gained from the tabulation in paragraph 786, it is possible to state the following rule which must be memorized: The characteristics of the logarithms of numbers between zero and one is negative, and numerically one greater than the number of zeros between the decimal point and the first nonzero digit. Example: The characteristic of log .0058 is (-3), that of log .68 is (-1), and that of log .00003792 is (—5). Find log .0068. Its characteristic is (−3), the sequence of digits is 6, 8, 0, and therefore the mantissa is .8325. Hence with the minus sign above the characteristic, which emphasizes that only the characteristic is negative, the mantissa always being positive. Another convenient way of writing negative characteristics is by adding and subtracting 10 to the characteristic found by the rule. Example: log .0068-7.8325-10. The mantissa is found, by interpolation if necessary, exactly as before. e. If given a logarithm with a negative characteristic and asked to find the number which has this for a logarithm, the decimal point is located by applying the foregoing rule. Examples: (1) Log N=8.6475-10, find N. The given mantissa is .6474 for which the sequence of digits of N is 4441. The given characteristic is (-2); then, by the rule, there is one zero between the decimal point and the first nonzero digit. Hence N=.0441 (2) If log S=9.6075-10, proceding in the same manner, one finds S=.405. f. Exercises. (1) Give the characteristics of the logarithms of the following numbers: .69, .0038, .6007, .00005, 65.3, 3.12, .00312. (2) Find the logs of the above numbers. 79. Multiplication and division.-a. General. It was found that logarithms are exponents, and hence the laws of operation upon logarithms are the same as those governing the combination of exponents. If a, m, and n are any three numbers whatsoever, two of these rules are: Thus, briefly stated, multiplication is accomplished by adding exponents, division by subtracting the exponent of the denominator from that of the numerator. Examples: (1) Multiply 60 by 3 using logarithms. It is found that log 60=1.7782, that is, 60=101.7782; and log 3.4771 or 3=10-4771. Hence, 60X3=101.778210.4771 — 102.2553. = Examples: (1) To perform divisions, subtract the log of the denom inator from the log of the numerator: Find N in N 200 40 If the log of the denominator is larger than that of the numerator, the subtraction will yield a negative number. Incorporate the negative part entirely into the characteristic, thus keeping the mantissa positive, by increasing and decreasing the log of the numerator by ten (or any appropriate number). 20 (2) Divide 20 by 30; write N, then log 20=1.3010=11.3010-10 Subtra tract log N= 9.8239-10 N= .6667 d. Exercises.-Perform the indicated operations: 80. Powers of numbers. a. Rule.-Powers of numbers are evaluated by applying to logarithms another law of exponents. This rule is that, for any numbers, a, m, and n, for example amn=am×n=amn or, briefly stated, powers are obtained by multiplying the exponents. For example: (72)3-76 b. Example: (1) Find 63. Say 63=N, then since log 6-.7782, the problem becomes (10-7782)3=N. Applying the aforegoing law for exponents, one gets 102.3346=N, and N=216. This process is written more briefly in the following form: 81. Extraction of roots.-a. First, express an indicated root of a number as an exponent with the use of fractional exponents. Hence 1 ana; the indicated root appears as the denominator of the fractional exponent. Example: √7=71/2 .29=(.29)1/5 162=(162)1/7=162/7 b. To find a root of a number, write the number with appropriate fractional exponent, and then proceed as in the preceding section. Example: (1) Find the square root of five. |