If the number is less than unity, but not an integral power of 10, the characteristic is positive, and is equal to the number of ciphers between the decimal point and the first significant digit. If the number is greater than unity, but not an integral power of 10, the characteristic is negative, and is equal to the number of digits to the left of the decimal point. The illog of a number is another number of which the given number is the log. Thus, if A=log B, then B=illog A. The table of illogs is used for finding results, as it will be remembered that the sum of the logs of two numbers, for example, is not the product of the two numbers, but the log of the product. The required product will therefore be the illog of the sum of the two logs. This table is used in almost the same way as the preceding tables, with the very important difference that the mantissa alone is to be used for entering the table, and the characteristic is only to be used to fix the position of the decimal point, the rules for which are the same as for logs. As stated on each page of this table, the proportional parts are to be added. It has been shown above that the four arithmetical processes of multiplication, division, involution, and evolution can be simplified by the use of logs. It is more convenient, however, to use logs and cologs for division; and other functions, yet to be described, for involution and evolution. Some examples of multiplication and division are given below :— To multiply 1.6732 by 42.363, If it is required to evaluate C", it has been shown above that log (C")=n × log C. If n is a simple number, it is not very troublesome to multiply the value of log C taken from the tables by n; but if, as often happens, n is a number of several digits, it will be more economical of time to carry out the multiplication by means of logs. Taking logs once more : log {log (C")}=log n+log (log C). The log of the log of a number is called its lolog, so that the above relation may be written thus : : By such procedure, the processes of involution and evolution are reduced to addition, but two separate references to the tables are required for each lolog. be saved by using a table which gives the log of the log of a number using a table which gives the illog of the illog of a number directly. called lolog and illolog tables respectively. Much time can directly, also by These tables are The illolog of a number is the number of which the given number is the lolog. Thus, if A=lolog C, then C-illolog A. It is helpful to remember that : (1) The illog of the log of a number is the number itself. For the reason previously stated, the mantissae only are given in ordinary log tables, the characteristic being supplied by inspection. For the same reason, a table of illogs is entered with the mantissa only; the characteristic merely determines the position of the decimal point. This simplicity is not possible in a table of lologs, as their mantissae depend on the position of the decimal point as well as on the actual digits. It follows, therefore, that a table of lologs must give the characteristics as well as the mantissae; and a table of illologs must be entered with both characteristic and mantissa. From this simple explanation, it might be inferred that any one accustomed to the use of ordinary logs might forthwith proceed to use a table of lologs for calculations requiring involution and evolution; but there is one difficulty which causes considerable trouble and confusion until it is appreciated and allowed for. The difficulty in question arises from the fact that the logs of numbers less than unity are negative; consequently it is necessary to provide logs of negative numbers if the lolog table is to be complete. In the true sense of the term, a negative number cannot have a log; but, fortunately, no difficulty need arise on this account, as the sign is an external feature which does not affect the numerical part, and it can be dealt with separately. This may be made clearer, perhaps, by pointing out that the numerical value of the product of each of the four following pairs of factors is the same : +2x+3=+6 +2x-3=-6 -2x-3+6 -2x+3=-6 It is clear that the product in each of the above cases could be found by adding log 2 to log 3, if the sign of the result were separately determined. Let it be required to evaluate (4.0)12 by ordinary logs. Then : Now, let it be required to evaluate (0-4)12 by ordinary logs. We should proceed thus: log 0.4 1.60206 This must now be multiplied by 1-2, which cannot be done in the ordinary way, owing to the fact that part of the number is negative, and part of it positive. The most convenient way of carrying out this multiplication is to subtract the mantissa from the characteristic, which leaves a purely negative number that can be multiplied by 1.2 in the ordinary way. Thus we have: The mantissa must now be made positive by adding 1 to it, which must be counterbalanced by subtracting 1 from the characteristic, then : Therefore log (0.4)12=1.52247 : illog 1.52247=0.33302 (0.4)12=0-33302 It will be seen that there is an important difference of procedure in these two examples. In the first example, the number to be involved is greater than unity. The result of this is that its log is positive, and consequently the product, after multiplying by 1-2, is positive also, so that the last mantissa is positive without the addition and subtraction of unity. In the second example, the number to be involved is less than unity. The result of this is that its log has a negative characteristic, which means that the log as a whole is negative. The product, after multiplying by 1.2, is therefore negative, so that addition and subtraction of unity are necessary before the number corresponding to the last log can be looked out in tables. From these two simple examples it will be seen that a certain amount of care is necessary to remember whether the logs are positive or negative. If the number to be involved is greater than unity, it is improbable that a mistake will be made; but if it is less than unity, experience shows that one is very liable to get confused. In the tables, the need for this care is entirely obviated by printing the lologs of numbers less than unity in red; whilst those of numbers greater than unity are printed in black. The red values are added and subtracted in the usual way, no notice being taken of the sign or colour until the result is being looked out, when, if a red lolog has been used, the result will be the illolog of a red number. With a little care it would be possible to remember whether the illolog of a red or black number was required; but most people will probably find it safer to use some visible reminder. If red ink is available, undoubtedly the simplest and safest thing to do is to write down red lologs in red ink, also the result after adding a log or a colog to a red lolog. Logs and cologs are, of course, always black. The rules for involution and evolution may now be stated : To raise a number to the nth power, add the log of n to the lolog of the number, and the illolog of this sum is the desired result. The sum is the same colour as the original lolog. To extract the nth root of a number, add the colog of n to the lolog of the number, and the illolog of this sum is the desired result. The sum is the same colour as the original lolog. The relation which exists between two lologs which are equal in magnitude, but of different colour, can be usefully employed. The easiest way to explain it is by considering an example, for which purpose a pair of reciprocal numbers is selected :- These two numbers, obviously, will have the same lolog, namely-1.77964; but, whereas lolog 4 is printed in black, lolog 0.25 is printed in red. to be evaluated with no more labour than that necessary to evaluate C". The rule is : When performing a process of involution or evolution corresponding to a negative index, merely change the colour of the lolog, and then proceed exactly as for a positive If the reader is accustomed to the use of log tables, or has followed the foregoing explanation, there is nothing further in the use of lologs and illologs or the arrangement of the tables that needs detailed description. It is, of course, impossible to understand and use lologs until one is accustomed to the use of logs. The following examples should be carefully studied and verified from the tables. This is, perhaps, the most direct method of becoming familiar with the small points in which these tables vary from the log tables : |