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 written over the characteristic, and not before it, to indicate t the characteristic alone is negative, and not the whole express 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 167. The logarithm of 5 and its powers can easily be obtained from log 2; for 10 2 log 5=log =log 10-log 2=1-log 2. Example. Find the value of the logarithm of the reciprocal of 324 5/125, having given log 2=3010300, log 3=4771213. EXAMPLES. XIV. a. 1. Find the logarithms respectively of the numbers 1024, 81, 125, 01, 3, 100, to the bases 2, √3, 4, 001, 1, 01. 2. Find the values of log, 16, log81 243, log.01 10, log49 343.√7. 3. Find the numbers whose logarithms respectively to the bases 49, 25, 03, 1, 64, 100, ·1, of the logarithms of 325, 1603, 2400, 10000, 19, of 5. 3, 11, 7, 9, 21. Write down the characteristics of the common logarithms 3.26, 523·1, 03, 1.5, 0002, 3000·1, 1. 6. The mantissa of log 64439 is 8091488, write down the logarithms of '64439, 6443900, 00064439. 7. The logarithm of 32.5 is 15118834, write down the numbers whose logarithms are 5118834, 2-5118834, 45118834. [When required the following logarithms may be used 19. Find the seventh root of 7, given log 1.320469=1207283. 20. Find the cube root of '00001764, given log 260315=5·4154995. 21. Given log 3571=3·5527899, find the logarithm of 3.571 × 03571 × 3571. 22. Given log 11=1·0413927, find the logarithm of (-00011)× (1-21)2 × (13·31)÷12100000. 23. Find the number of digits in the integral parts of 24. How many positive integers have characteristic 3 when the base is 7? 168. Suppose that we have a table of logarithms of numbers to base a and require to find the logarithms to base b. Let N be one of the numbers, then log, N is required. Now since N and b are given, loga N and logɑb are known from the Tables, and thus log, N may be found. Hence it appears that to transform logarithms from base a 1 loga b this is a to base b we have only to multiply them all by constant quantity and is given by the Tables; it is known as the modulus. If in equation (1) we put a for N, we obtain 169. The following examples further illustrate the great use of logarithms in arithmetical work. Example 1. Given log 2=3010300 and log 4844544=6.6852530, find the value of (6-4)1 × (+/-256)3÷√/80. Example 2. Find how many ciphers there are between the decimal point and the first significant digit in (0504)10; having given log 2=301, log 3=477, log 7=⚫845. Denote the expression by E; then |