Hence a difference in the fourth figure of 1 2 3 4 5 6 7 8 9 produces a difference in the logarithm of 71.4 2.1 2.8 3.54.24.95.66.3 or taking the nearest whole numbers of 1 1 2 3 3 4 5 6 6 Hence these small additions to the logarithms are spoken of as differences, they are nearly though not exactly proportional to the differences between the numbers. (32) TO FIND A NUMBER FROM ITS Neglecting the characteristic, find in the tables the mantissa next below that of the given logarithm, take for the first two figures of the number those at the left hand of the row in which the mantissa stands, and for the third figure of the number that at the top of the column in which the mantissa stands. To find the fourth figure of the number subtract the mantissa found in the tables from that of the given logarithm, and look along the row for the next smaller difference; the figure standing at the top of that column of differences is the required fourth figure of the number. Reckon the position of the decimal point from the characteristic. Thus *log1 80626.4, log 1.814965.3, log4-7782-0006, log1 00121·003. log1 2.8095 644.9, log1 3.7821 = .006054 89 = 6 diff. for 9. 18 3 diff. for 4. It must be remembered that the fourth figure obtained by four-figure logarithms may be too great or too small by 1. * log-1 means find the number from its logarithm. (33) THE USE OF LOGARITHMS. The four following rules give the chief uses of logarithms. (i) To multiply numbers add their logarithins, and the sum is the logarithm of the product. (ii) To divide numbers subtract the sum of the logarithms of the divisors from the sum of the logarithms of the dividends, the difference is the logarithm of the quotient. (iii) To find the power of a number multiply the logarithm of the number by the index of the power to which it is to be raised, and the product is the logarithm of the required power. (iv) To find the root of a number, divide the logarithm of the given number by the index of the root which is to be extracted, the quotient is the logarithm of the required root. Thus 12340 log 12340 = 4·0913, log-1 4.0913 =log11·3628 = 23.06. If the root of a decimal, the logarithm of which has a negative characteristic, has to be found, a number must be added both to the characteristic and mantissa such as to make the characteristic an exact multiple of the index of the root by which it is to be divided. The root is then found by the method given above. Thus 007654 log 0076543-8839; add 1 to each. 4+1.8839 = 14709. log1I-4709-29575. Again√log 2-log 3-1-8239; add 1 to each. 2+1.8239 =Ï·9119. log ̄1Ï·9119·8164. (34) EXAMPLES OF THE USE OF LOGARITHMS IN SOLVING CHEMICAL PROBLEMS. It is evident that in complicated cases, after all signs of addition and subtraction have been got rid of, the results are more easily obtained by logarithmic than by common arithmetic. It is worth while to remember the logarithms of a few numbers which constantly occur, such as = log 3.14159 4971, log 22.32 = 1·3487, = 2.8808, log 0896 2.9523. = A few examples of different kinds of problems will now be given: (i) The expansion of gases by heat. It was pointed out at the end of Section. 16, that the expansion of a gas may be represented by the formula where 00367 is the coefficient of expansion of the gas which is usually expressed by a. If a table of the logarithms of (1+00367) has been prepared for the different values of t, it is easy to calculate the change in volume produced by any change in temperature. Cf. Table VII. Thus, to find the new volume if 1000 c.c. of gas are heated from 0° C. to 47° C. If differences are given in these tables they are usually given for 1° C. Again 1426 c. c. of gas are heated from 40° C. to 50o C., find the new volume. (ii) The diffusion of gases is treated with great facility by logarithms. If 436 c. c. of hydrogen chloride (HCl = 36.5) diffuse through a certain apparatus in a certain time, what volume |