Thus in arithmetical calculations in which 10 is the base, we usually write log 2, log 3,...... instead of log10 2, log10 3,...... Logarithms to the base 10 are known as Common Logarithms; this system was first introduced in 1615 by Briggs, a contemporary of Napier the inventor of Logarithms. Before discussing the properties of common logarithms we shall prove some general propositions which are true for all logarithms independently of any particular base. 154. The logarithm of 1 is 0. For ao=1 for all values of a; therefore log 1=0, whatever the base may be. 155. The logarithm of the base itself is 1. 156. To find the logarithm of a product. Let MN be the product; let a be the base of the system, and suppose so that x=loga M, a*=M, y=loga N; a2 = N. Thus the product MN=a* xav=a¤+v; whence, by definition, log, MN=x+y =loga M+loga N. Similarly, loga MNP-loga M+loga N+loga P; and so on for any number of factors. Example. log 42=log (2 × 3 × 7)=log 2+ log 3+log 7. 157. To find the logarithm of a fraction. 158. To find the logarithm of a number raised to any power, integral or fractional. Let loga (MP) be required, and suppose X= =loga M, so that a2=M; MP=(a*)P=αpx; then 159. It follows from the results we have proved that (1) the logarithm of a product is equal to the sum of the logarithms of its factors; (2) the logarithm of a fraction is equal to the logarithm of the numerator diminished by the logarithm of the denominator; (3) the logarithm of the pth power of a number is the logarithm of the number; 1 p times (4) the logarithm of the rth root of a number is of the logarithm of the number. Thus by the use of logarithms the operations of multiplication and division may be replaced by those of addition and subtraction; the operations of involution and evolution by those of multiplication and division. 160. From the equation 10= N, it is evident that common logarithms will not in general be integral, and that they will not always be positive. For instance Again, 3154> 103 and <104; .. log 3154=3+a fraction. '06>10-2 and <10-1; .. log 062+ a fraction. 161. DEFINITION. The integral part of a logarithm is called the characteristic, and the decimal part is called the mantissa. The characteristic of the logarithm of any number to the base 10 can be found by inspection, as we shall now shew. 162. To determine the characteristic of the logarithm of any number greater than unity. It is clear that a number with two digits in its integral part lies between 101 and 102; a number with three digits in its integral part lies between 102 and 103; and so on. Hence a number with n digits in its integral part lies between 10–1 and 10". Let N be a number whose integral part contains n digits; then N=10(n-1)+a fraction .'. log N=(n − 1)+a fraction. Hence the characteristic is n- -1; that is, the characteristic of the logarithm of a number greater than unity is less by one than the number of digits in its integral part, and is positive. 163. To determine the characteristic of the logarithm of a decimal fraction. A decimal with one cipher immediately after the decimal point, such as 0324, being greater than 01 and less than 1, lies between 10-2 and 10-1; a number with two ciphers after the decimal point lies between 10-3 and 10-2; and so on. Hence a decimal fraction with n ciphers immediately after the decimal point lies between 10-(n+1) and 10-". Let D be a decimal beginning with n ciphers; then Hence the characteristic is (n+1); that is, the characteristic of the logarithm of a decimal fraction is greater by unity than the number of ciphers immediately after the decimal point and is negative. 164. The logarithms to base 10 of all integers from 1 to 200000 have been found and tabulated;, in most Tables they are given to seven places of decimals. The base 10 is chosen on account of two great advantages. (1) From the results already proved it is evident that the characteristics can be written down by inspection, so that only the mantissæ have to be registered in the Tables. (2) The mantissæ are the same for the logarithms of all numbers which have the same significant digits; so that it is sufficient to tabulate the mantissæ of the logarithms of integers. This proposition we proceed to prove. 165. Let N be any number, then since multiplying or dividing by a power of 10 merely alters the position of the decimal point without changing the sequence of figures, it follows that N× 10o, and N÷10%, where p and q are any integers, are numbers whose significant digits are the same as those of N. In (1) an integer is added to log N, and in (2) an integer is subtracted from log N; that is, the mantissa or decimal portion of the logarithm remains unaltered. In this and the three preceding articles the mantissæ have been supposed positive. In order to secure the advantages of Briggs' system, we arrange our work so as always to keep the mantissa positive, so that when the mantissa of any logarithm has been taken from the Tables the characteristic is prefixed with its appropriate sign, according to the rules already given. 166. In the case of a negative logarithm the minus sign is written over the characteristic, and not before it, to indicate that the characteristic alone is negative, and not the whole expression. Thus 4-30103, the logarithm of '0002, is equivalent to −4+·30103, and must be distinguished from -4.30103, an expression in which both the integer and the decimal are negative. In working with negative logarithms an arithmetical artifice will sometimes be necessary in order to make the mantissa positive. For instance, a result such as -3.69897, in which the whole expression is negative, may be transformed by subtracting 1 from the integral part and adding 1 to the decimal part. Thus −3·69897= −4+(1 − ·69897)=4·30103. Example 1. Required the logarithms of 0002432. In the Tables we find that 3859636 is the mantissa of log 2432 (the decimal point as well as the characteristic being omitted); and, by Art. 163, the characteristic of the logarithm of the given number is - 4; .. log 0002432= 4.3859636. Example 2. Find the cube root of ⚫0007, having given log 7=-8450980, log 887904=5.9483660. Let x be the required cube root; then that is, but 1 1 log = log('0007) = (4-8450980) = (6+2·8450980); 3 3 log x= 2-9483660; 3 log 887904-5.9483660; .. x=0887904. 167. The logarithm of 5 and its powers can easily be obtained from log 2; for log 5= =log 10 Example. Find the value of the logarithm of the reciprocal of 324 5/125, having given log 2=3010300, log 3='4771213. |