MULTIPLICATION BY LOGARITHMS. 11. When it is required to multiply numbers by means of their logarithms, we first find from the table the loga rithms of the numbers to be multiplied; we next add these logarithms together, and their sum is the logarithm of the product of the numbers (Art. 3). The term sum is to be understood in its algebraic sense; therefore, if any of the logarithms have negative characteristics, the difference between their sum and that of the positive characteristics, is to be taken; the sign of the remainder is that of the greater sum. EXAMPLES. 1. Multiply 23.14 by 5.062. log 23.141.364363 log 5.0620.704322 Product, 117.1347 . . . 2.068685 2. Multiply 3.902, 597.16, and 0.0314728 together. Here, the 2 cancels the + 2, and the 1 carried from the decimal part is set down. 3. Multiply 3.586, 2.1046, 0.8372, and 0.0294 together. In this example the 2, carried from the decimal part, cancels 2, and there remains I to be set down. DIVISION OF NUMBERS BY LOGARITHMS. 12. When it is required to divide numbers by means of their logarithms, we have only to recollect, that the subtraction of logarithms corresponds to the division of their numbers (Art. 4). Hence, if we find the logarithm of the dividend, and from it subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient. This additional caution may be added. The difference of the logarithms, as here used, means the algebraic differ ence; so that, if the logarithm of the divisor have a nega tive characteristic, its sign must be changed to positive, after diminishing it by the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of the dividend is nega tive, it must be treated as a negative number. 2. To divide 0.06314 by .007241. log 0.06314= 2.8003(5 Here, 1 carried from the decimal part to the 3, changes it to 2, which being taken from 2, leaves 0 for the cha racteristic. Here, the 1 taken from I, gives 2 for a result, as set down. ARITHMETICAL COMPLEMENT. 13. The Arithmetical complement of a logarithm is the number which remains after subtracting the logarithm from 10. Thus, of 9.274687. 10-9.274687=0.725313. 0.725313 is the arithmetical complement 14. We will now show that, the difference between two logarithms is truly found, by adding to the first logarithm the arithmetical complement of the logarithm to be subtracted, and then diminishing the sum by 10. and Let a = the first logarithm, b=the logarithm to be subtracted, c=10-b= the arithmetical complement of b. Now the difference between the two logarithms will be expressed by a-b. But, from the equation c=10-b, we have c-10=-b, hence, if we place for -b its value, we shall have a-b=a+c-10, which agrees with the enunciation. When we wish the arithmetical complement of a logarithm, we may write it directly from the table, by subtract ing the left hand figure from 9, then proceeding to the right, subtract each figure from 9 till we reach the last figure, which must be taken from 10: this will be the same as taking the logarithm from 10. Hence, to perform division by means of the arithmetical complement, we have the following RULE. To the logarithm of the divr lend add the arithmetical com plement of the logarithm of the divisor: the sum, after sub tracting 10, will be the logarithm of the quotient. In this example, the sum of the characteristics is 8, from which, taking 10, the remainder is 2. FINDING THE POWERS AND ROOTS OF NUMBERS BY LOGARITHMS. 15. We have (Art. 3), 10m = M. Raising both members of this equation to the nth power, we have, 10TM×"=M", in which mXn is the logarithm of M" (Art. 1): hence, The logarithm of any power of a given number is equal to the logarithm of the number multiplied by the exponent of the and extracting the nth root of both members, we have m 10" = in which is the logarithm of M: that is, n The logarithm of the root of a given number is equal to the logarithm of the number divided by the index of the root. EXAMPLES. 1. What is the 5th power of 9? = Log 9 0.954243; 0.954243 x 5= 4.771215; whole number answering to 4.771215 is 59049. 2. What is the 7th power of 8? 3. What is the cube root of 4096? Ans. 2097152. Log 4096 = 3.612360; 3.612360÷ 3 = 1.204120; number answering to 1.204120 is 16. But, and, 4. What is the 4th root of .00000081? Log .00000081 = 7.908485; 7.9084858 +1.908485; 8+1.908485 ÷ 4 = 2.477121, the number answering to which is .03, which is the root. When the characteristic of the logarithm is negative, and not divisible by the index of the root, add to it such a negative number as will make the sum exactly divisible by the index, and then prefix the same number to the first decimal figure of the logarithm. 5. What is the 6th root of .0432? Ans. .592353 + 6. What is the 7th root of .0004967? Ans. .3372969 |