CHAPTER VIII. THE NATURE AND PROPERTIES OF LOGARITHMS. 182. DEF. 1. Logarithms are a series of magnitudes increasing by a common Difference, corresponding to another series of magnitudes increasing by a common Multiplier: thus, if the former series be the natural numbers increasing by the common difference 1, as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, &c.; and the latter begin with 1 and increase by the common multiplier or factor 2, as 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, &c. or, 2o, 2', 22, 23, 24, 25, 2o, 2′, 2o, 2o, &c.; any term of the former series is defined to be the logarithm of the corresponding term of the latter: thus, we have where the number 2, which has been arbitrarily assumed, is called the Radix or Base of the System of Logarithms: and it is evident that if the magnitude of any term in either of these series of quantities be assigned, that of the corresponding term in the other will be given. Also, if an arithmetic mean between any two of the terms of the former series be found, it is manifest from the manner in which the two series are connected, that a geometric mean between the two corresponding terms of the second series must have the same relation to it, throughout the whole extent of both the series adopted. A simpler idea of these numbers will perhaps be had by defining the logarithm of a magnitude to be the index of a fixed number which, when raised to the power denoted by that index, produces the magnitude, the fixed THE NATURE AND PROPERTIES OF LOGARITHMS. 181 number being assumed of any magnitude whatever, that of unity excepted because every power of 1 is 1. 183. DEF. 2. If the number 10, which is the Base of the Common System of Notation, be adopted for the base of the logarithms as above defined, the terms of the series 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, &c. will, by the last Article, be the logarithms of the corresponding terms of the series. 10°, 101, 10, 103, 10, 105, 10, 107, 108, 10°, &c.: that is, in a system of logarithms whose base is 10, 0 = log 10° or 1; 9= log 10° or 1000000000; &c. = &c.: and it is further manifest from what has been said, that the arithmetic mean between any two terms of the first series will be the logarithm of the geometric mean between the two corresponding terms of the second. The arithmetic mean of 0 and 1 is .5: the geometric mean of 1 and 10 is 3.16227 &c.; and therefore .5= the logarithm of 3.16227 &c. The arithmetic mean of .5 and 1 is .75: the geometric mean of 3.16227 &c. and 10 is 5.62341 &c.; whence .75 the logarithm of 5.62341 &c. = The arithmetic mean of 1 and 2 is 1.5: the geometric mean of 10 and 100 is 31.62277 &c.; whence 1.5= the logarithm of 31.62277 &c.: and by continued repetitions of the process upon these and other numbers it follows that the logarithms of all magnitudes whatever might be ascertained, though the labour requisite to do it would be immense. It appears that O is the logarithm of 1 in any system whatever its base may be. 184. DEF. 3. There is no difficulty in seeing that the logarithm of a magnitude between 1 and 10 will be a decimal fraction: that of a magnitude between 10 and 100 will be 1 with a decimal fraction annexed: that of one between 100 and 1000 will be 2 with a corresponding decimal fraction, and so on: for, and the integers 0, 1, 2, 3, &c., to the left of the decimal points in the logarithms of magnitudes are called the Characteristics of those logarithms: thus, 0 is the characteristic of the logarithms of all magnitudes between 1 and 10; 1 is the characteristic of the logarithms of all magnitudes between 10 and 100; 2 that of all magnitudes between 100 and 1000; &c. 185. DEF. 4. If the logarithms of all magnitudes be calculated by processes analogous to the one above explained, (or indeed by any other methods which the present advanced state of mathematical science may suggest, but which were unknown to the more early writers upon the subject,) and the results be put into the form of a table, we shall have what is called a Table of Logarithms; and this may be used to facilitate the arithmetical operations of Multiplication, Division, Involution and Evolution, and to render these operations when applied to surds or other complicated magnitudes, exceedingly concise and easy. The advantages thus conferred upon the practical mathematician will be fully explained and exemplified in the following Articles. 186. The Logarithm of the Product of two magnitudes is equal to the sum of the Logarithms of those magnitudes. Resuming the two series of magnitudes last used, we logarithms, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, &c.: numbers, 1, 10, 102, 103, 10, 105, 10°, 107, 10°, 10°, &c.: and in these we observe that log (1 × 10) = log 10=1=0+1 log (10 × 100) = log 1000 = 3 = 1 + 2 log (10 × 1000) = log 10000 = 4=1+3 log (100 x 10000) = log 1000000 = 6 = 2 +4 also, it is manifest from the formation of these numbers, that the same must be universally true, and that log 6 = log (2 × 3) = log 2 + log 3: log 24 = log (4 × 6) = log 4 + log 6, &c. : and this property may be rendered available to facilitate the multiplication of numbers whenever a table of logarithms, as explained in the last Article, is at hand. Ex. Let it be required to find the product of the numbers 7 and 23, by means of a table of logarithms. Here, referring to tables of this description, we find log 7=0.8450980, log 23 = 1.3617278, the characteristics which are there omitted being 0 and 1 respectively, for the reasons assigned in Article (184): whence, the logarithm of the required product will be 0.8450980+ 1.3617278 2.2068258; = and by looking again into the table, we find that this quantity without the characteristic, namely, .2068258 is the logarithm of 161, the characteristic itself merely shewing that the number is between 100 and 1000: that is, we have now the logarithm of the required product equal to the logarithm of 161, and consequently the product will be 161. The operation above given may be more conveniently arranged as follows: and therefore 7 × 23 = 161, as we know to be the case. Precisely in the same manner, whatever be the number of factors as 17, 26, 35, &c., we shall have log (17 × 26 × 35 × &c.) = log 17 + log (26 × 35 × &c.) = log 17+log 26+ log 35+ &c., from which the product may be ascertained as in the preceding example. 187. The Logarithm of the Quotient of two magnitudes is equal to the difference of the Logarithms of those mag nitudes. Referring to the statement made at the head of the last Article, we see that log (10+1)= log 10=1=1-0 = log 10- log 1: log (1000+10)= log 100=2=3-1 = log 1000-log 10: log (1000000100) = log 10000=4=6-2 = log 1000000-log 100: &c.: and the general nature of these operations leads us to conclude similarly, that log 3 = log (62) = log 6 - log 2: log 9 = log (27 ÷ 3) = log 27 - log 3: log 23 = log (161÷7) = log 161 - log 7: &c. This property will enable us to ascertain the quotient of two quantities, by the help of a logarithmic table. Ex. What is the quotient arising from the division of 324 by 27? Here, log (324÷27) = log 324 - log 27 = 2.5105452 – 1.4313639 = = 1.0791813 = log 12: |