by x, x', x', x''', &c. Then by definition (Art. 257), we have a*=y, a*'=y', a*"'=y', a='"'=y'" . . . &c. Multiplying these equations together, member by member, and applying the rule for the exponents, we have x+x′+x"+x"". = log. yy'y'y", that is, the sum of the logarithms of any number of factors is equal to the logarithm of the product of those factors. 259. Suppose it were required to divide one number by another. Let y and y denote the numbers, and x and x their logarithms. We have the equations that is, the difference between the logarithm of the dividend and the logarithm of the divisor, is equal to the logarithm of the quotient. Consequences of these properties. A multiplication can be performed by taking the logarithms of the two factors from the tables, and adding them together; this will give the logarithm of the product. Then finding this new logarithm in the tables, and taking the number which corresponds to it, we shall obtain the required product. Therefore, by a simple addition, we find the result of a multiplication. In like manner, when one number is to be divided by another, subtract the logarithm of the divisor from that of the dividend, then find the number corresponding to this difference; this will be the required quotient. Therefore, by a simple subtraction, we obtain the quotient of a division. Formation of Powers and Extraction of Roots. 260. Let it be required to raise a number y to any power de If a denotes the base of the system, and x the loga that is, if the logarithm of any number be multiplied by the exponent of the power to which the number is to be raised, the product will be equal to the logarithm of that power. As a particular case, take n= =1; there will result m. log y= logy; an equation which is susceptible of the above enunciation. 261. Suppose, in the first equation, m=1; there will result that is, the logarithm of any root of a number is obtained by divid ing the logarithm of the number by the index of the root. Consequence. To form any power of a number, take the logarithm of this number from the tables, multiply it by the exponent of the power; then the number corresponding to this product will be the required power. In like manner, to extract the root of a number, divide the loga. rithm of the proposed number by the index of the root, then the number corresponding to the quotient will be the required root. Therefore, by a simple multiplication, we can raise a quantity to a power, and extract its root by a simple division. 263. The properties just demonstrated are independent of any system of logarithms; but the consequences which have been deduced from them, that is, the use that may be made of them in numerical calculations, supposes the construction of a table, containing all the numbers in one column, and the logarithms of these numbers in another, calculated from a given base. Now, in calculating this table, it is necessary, in considering the equation aa=y, to make y pass through all possible states of magnitude, and determine the value of a corresponding to each of the values of y, by the method of Art. 265. The tables in common use, are those of which the base is 10 and their construction is reduced to the resolution of the equation 10=y. Making in this equation, y successively equal to the series of natural numbers, 1, 2, 3, 4, 5, 6, 7 we have to resolve the equations 101, 102, 103, 10=4... We will moreover observe, that it is only necessary to calculate directly, by the method of Art. 255, the logarithms of the prime numbers 1, 2, 3, 5, 7, 11, 13, 17 .; for as all the other entire numbers result from the multiplication of these factors, their logarithms may be obtained by the addition of the logarithms of the prime numbers (Art. 258). Thus, since 6 can be decomposed into 2×3, we have log 6 log 2+ log 3; in like manner, 24-23×3; hence log 24=3 log 2+ log 3. Again, 360=23×32X5; hence log 360 3 log 2+2 log 3+ log 5. It is only necessary to place the logarithms of the entire numbers in the tables; for, by the property of division (Art. 259), we obtain the logarithm of a fraction by subtracting the logarithm of the divisor from that of the dividend. 263. Resuming the equation 10*=y, if we make x=0, 1, 2, 3, 4, we have 5, n-1, n. y=1, 10, 100, 1000, 10000, 100000,... And making 10%~, 10". From which we see that, the logarithm of a whole number will become the logarithm of a corresponding decimal by changing its sign from plus to minus. 264. Resume the equation a*=y, in which we will first suppose a>1. Then, if we make y=1 we shall have If now, y diminishes a will increase, and when y becomes 0, we 1 have a -- 0 or a= ∞ (Art. 112); but no finite power of a απ is infinite, hence x = ∞ and therefore, the logarithm of 0 in a sys. tem of which the base is greater than unity, is an infinite number and negative. 265. Again take the equation a*=y, and suppose the base a<1. Then making, as before, y=1, we have ao=1. If we make y less than 1 we shall have a*=y<1. Now, if we diminish y, x will increase; for since a<1 its powers will diminish as the exponent x increases, and when y=0, x must be infinite, for no finite power of a fraction is 0. Hence, the loga. rithm of 0 in a system of which the base is less than unity, is an infinite number, and positive. Logarithmic and Exponential Series. 266. The method of resolving the equation a=b, explained in Art. 255, is sufficient to give an idea of the construction of logarithmic tables; but this method is very laborious when we wish to approximate very near the value of x. Analysts have discovered much more expeditious methods for constructing new tables, or for verifying those already calculated. These methods consist in the development of logarithms into series. Taking again the equation a*=y, it is proposed to develop the logarithm of y into a series involving the powers of y, and co-effi. cients independent of y. It is evident, that the same number y will have a different loga. rithm in different systems; hence the log y, will depend for its value, 1st. on the value of y; and 2dly, on a, the base of the system of logarithms. Hence the development must contain y, or some quantity dependent on it, and some quantity dependent on the base a. To find the form of this development, we will assume log y=A+By+Cy2+Dy3+, &c., in which A, B, C, &c. are independent of y, and dependent on the base a. Now, if we make y=0, the log y becomes infinite, and is either negative or positive according as the base a is greater or less than unity (Arts. 264 & 265). But the second member under this supposition, reduces to A, a finite number : hence the development cannot be made under that form. Again, assume log y=Ay+By+Cy3+Dy1+, &c. If we make y=0, we have log y∞ =0, |