SECTION 6. Meaning of Logarithms. Rules. Arrangement of Tables in common use. Method of taking out Logarithms, and Numbers to Logarithms. or 10100000 or 10 or 10 100000 0/1030103 So that we may get a result as near to 2 as we please; that is, we may find a decimal fraction x, which shall, as nearly as we please, satisfy the equation 102. The answer is x=30103 nearly. And in the same way we may find an approximate logarithm for any other number or fraction. These approximate logarithms are arranged in tables, with certain modifications derived from the following fundamental 2 X 5 10 Log. 2 Log. 5 = Log. 10 or 10 .3 ⚫301 •30103 = 1 9952623150 nearly. = 1.9998618696 nearly. =20000000200 nearly. rules, which are proved in works on the subject. 1. The logarithm of a product must be the sum of the logarithms of the factors. Thus, 6, 8, and 10, multiplied together, give 480; the logarithms of 6, 8, and 10, added together, give the logarithm of 480. The following instances may be immediately verified from any tables: 2. To find the logarithm of a quotient, subtract the logarithm of the divisor from the logarithm of the dividend. Thus, 20 divided by 5 gives 4; the logarithm of 20, diminished by the logarithm of 5, is the logarithm of 4: 3. The logarithm of a power, root, or combination of power and root, which is The reader must recollect throughout, that we here lay down rules only, not demonstrations. D denoted in algebra by a fractional exponent, is found by multiplying the loga rithm of the number given by the exponent in question. The following equations will set this in a clearer light: Log. aa or Log. a2 2 Log. a. 3 Log. a. 5. As the number increases the logarithm increases, and the greater the number the greater the logarithm: but the rate at which the logarithm increases is perpetually diminishing as the number increases. Thus we see that, as the number passes from 10,000 to 100,000 (through ninety thousand units) the logarithm passes from 4 to 5, receiving no greater increase than takes place while the number passes from 1 to 10 (through nine units only). 6. In any logarithm (4 6183 for instance) the whole numbev (4) is called the characteristic, and the remainder (6183) the decimal part of the logarithm. m m = Log. a. n 8. A fraction less than unity (5 for instance) has none but a negative logarithm: but that students may use logarithms who have not studied algebra, we affix a meaning to the term negative, for this subject only. The term multiplication is extended in arithmetic to whole numbers and fractions, so that multiplication, in its extended meaning, includes the first meaning of division: thus, to multiply by is to divide by 10. But from the connection which exists between multiplication of numbers and addition of logarithms, and also between division of numbers and subtraction of logarithms, we cannot use the word multiplication in an extended sense, which includes division, and keep rules (1) and (2) at the same time,* unless we also use the word addition in an extended sense, which includes subtraction. And this is done as follows: by 1 we mean a unit, with a warning, that in all operations performed upon this 1, we are to subtract where we should have added if the bar had been absent, and to add where we should have subtracted. And with this we say, that 1 being the logarithm of 10, I is the logarithm of 2 being the logarithm of 100, 2 is the logarithm of 100 the following are instances of the use of this sign, with the corresponding real operations:Multiply Divide 1000 by Log. 1000 Log. 1 1 1000 by 10 Log. 1000 = 3 10 = 3 = 1 10 7. In any number (368 414 for instance) the figures which precede the decimal point (the 3, the 6, and the 8,) are called integers, and those which follow the point are called decimals. And figures, when opposed to ciphers, are called significant. Thus, in 864000, 4 is the last significant figure; in 000193, 1 is the first significant figure. * The choice is, between making two rules, and using the words of one rule in a sense which will make that one include both. The latter is the more difficult at first, but the more convenient in the end. = or 100 = 5 1 100,000 2 + 3 - = 10 2 or 2 + 2 100= 10,000 3 2 = Log. 5 = Subtract 2.30103 which is log. 200} Ans. 200. But the necessity of using decimal places with a negative sign, can always be avoided, and the characteristic only made negative, as follows: for log 5 5 or log. or log. 5 log. 10 or ⚫69897 - 1 write ·69897 + ī Log. 5 = 169897 Add 1*0000 Ans. 10. Here the 1, which is carried after adding 1, 6, and 3, (where we have placed an asterisk instead of a cipher to mark the place) instead of increasing the 1, destroys it. What is 100÷·5? Log. 100 = 2.00000 Subtract 2 30103 as before. To make any logarithm which is entirely negative, negative in the characteristic only, make that characteristic greater by 1, and subtract the decimal part from 1. Ι I 1 1 From 2 Take 3 5 To multiply a logarithm with a negative characteristic by a whole number, proceed in all respects as in common multiplication, except only in subtracting, instead of adding the figures which are to be carried, so soon as the characteristic comes to be multiplied. 4.1 4·6 ī.61 2.55 4 2.44 3 5.65 2 32.8 7.2 When the multiplier exceeds 12, and the better way is to omit the characthe process is not performed in one line, teristic altogether, at first, and subtract the product arising from it afterwards, as in the following multiplication of 2.136 by 15. 4 X 15: ⚫136 15 680 136 2.040 = 60 Subtract 58 040 To divide a logarithm with a negative characteristic by a whole number, begin by increasing the characteristic until it is divisible by the whole number, make the quotient a negative characteristic for the result, and use the augment which was found necessary, as if it had been a remainder. Thus, to divide 14 by 2, increase the first 1, and make it 2 (necessary augment, 1) and 2 being contained in 2 once, I is the characteristic of the quotient. Then, taking the augment 1, prefix it to the 4, giving 14, which contains 2 seven times. Therefore 17 is the quotient. We can now give a logarithm, by help of the tables, to any number or fraction, and can, by the above conventions, make the rules marked (1) (2) and (3) include all cases of logarithmic operations, by help of the following rules. (a) An alteration in the position of the decimal point, alters only the characteristic, and not the decimal part of the logarithm, if the significant figures remain the same: thus all the following numbers and fractions have the same decimal part in their logarithms, with different characteristics. and these are the only numbers to which logarithms can be exactly found; the decimal places of all others being approximations only. Tables of logarithms (generally) contain the decimal part of the logarithm, which is evidently all that is necessary, as the characteristic can be found by the preceding rule. Being approximations, they are more or less correct according to the greater or smaller number of places which they give. Modern tables never have fewer than four, or more than seven decimal places. The following is the rule by which the power of a table of logarithms is to be judged. The number of places of figures rived from any table of logarithms, is which may be obtained in a result dethe same as the number of decimals to which the logarithms are carried. But towards the end of the table, the last place thus obtained cannot always be depended upon within a unit. We shall proceed to the description of the arrangements of several tables, such as are most likely to fall in the reader's way. |