Ex. 7. log. (a'/a')=log. a'+log. a2=3 log. a+log. a 4 15 = log. a 4 Ex. 8. log. Va3 — x3)m = m = n == log. (a-x)+ log.(a2+ax+x2) n {log. (a―x)+log. (a+x+2)+log. (a+x—z)} where z-ax Ex. 9. log. √a2+x2 = {{log. (a+x+2)+log. (a+x—2)}, 1o. If a>1, making x=0, we have N=1; the hypothesis x=1 gives N=a. As x passes from 0 up to 1, and from 1 up infinity, N will increase from 1 up to a, and from a up to infinity; so that being supposed to pass through all intermediate values, according to the law of continuity, N increases also, but with much greater rapidity. If we attribute negative values to x. we have N=ax , or N= Here, as x increas 1 ax. es, N_diminishes, so that x being supposed to increase negatively, N will decrease from 1 towards 0, the hypothesis = gives N=0. 2o. If a<1, put a= 1 1 b' where b>1, and we shall then have N= tive values to x. We here arrive at the same conclusion as in the former case, with this difference, that when x is positive N<1, and when x is negative N>1. bx or N=bx, according as we attribute positive or nega 3°. If a 1, then N=1. whatever may be the value x. = From this it appears, that, 1. In every system of logarithms the logarithm of 1 is 0, and the logarithm of the base is 1. II. If the base be >1, the logarithms of numbers >1 are positive, and the logarithms of numbers<1 are negative. The contrary takes place if the base be<1. III. The base being fixed, any number has only one real logarithm; but the same number has manifestly a different logarithm for each value of the base, so that every number has an infinite number of real logarithms. Thus, since 9'=81, and 3'=81, 2 and 4 are the logarithms of the same number 81, according as the base is 9 or 3. IV. Negative numbers have no real logarithms, for attributing to x all values from ― up to+∞, we find that the corres ponding values of N are positive numbers only, from 0 up to +∞. The formation of a table of logarithms consists in determining and registering the values of x which correspond to N=1, 2, 3,. in the equation, N=a* If we suppose m=a", making x=0, α, 2a, 3a, 4a, 5α, logarithms. y=1, m, m2, m3, m2, m3, the logarithms increase in arithmetical progression, while the numbers increase in geometrical progression; 0 and 1 being the first terms of the corresponding series, and the arbitrary numbers & and m the common difference and the common ratio. We may, therefore, consider the systems of values of x and y, which satisfy the equation N=a*, as ranged in these two progressions. α In order to solve the equation c=ax where c and a are given, and where x is unknown, we equate the logarithms of the two members, which gives us log. P-log. Q x= log. a If we have an equation a2=b, where z depends upon an unknown quantity x, and we have Since z= log. b log. a K some known number, the problem depends upon the solution of the equation of the nth degree. an equation of the second degree, from which we find x = x = 3. To find the value of x from the equation ᄆ x = cmx fx-p Taking the logarithms of each member, Or, (n) log. b=m x log. c+ (x —p) log. ƒ 2, (m log. c+log. ƒ) x2— (n log. b+p log. ƒ) x+a log. b=0 a quadratic equation, from which the value of x may be determined. In like manner, from the equation we find cmx abnx-1 Equations of this nature are called Exponential Equations. Let N and N + 1 be two consecutive numbers, the difference of their logarithms, taken in any system, will be log. (N+1)—log. N=log. (N+1) =log. (1+ a quantity which approaches to the logarithm of 1, or zero, in decreases, that is, as N increases. Hence it The difference of the logarithms of two consecutive numbers is less in proportion as the numbers themselves are greater.. When we have calculated a table of logarithms for any base a, we can easily change the system, and calculate another table for a new base b. Let c=b, x is the log of c in the system whose base is b; Taking the logs. in the known system, whose base is a, we have 1 (A) hence log. bi this last factor 1 log. b The log. of c in the system whose base is b, is the quotient arising from dividing the log. of c by the log. of the new base b, both these last logs. being taken in the system whose base is a. In order to have x the log. of c in the new system, we must multiply log. c by is constant for all numbers, and is called the Modulus; that is to say, if we divide the logs. of the same number c taken in two systems, the quotient will be invariable for these systems, whatever may be the value of c, and will be the modulus, the constant multiplier which reduces the first system of logs. to the second. If we find it inconvenient to make use of a log. calculated to the base 10, we can in this manner, by aid of a set of tables calculated to the base 10, discover the logarithm of the given number in any required system. For example, let it be required, by aid of Briggs' tables, to 2 find the log. of in a system whose base is 3 Let x be the log. sought, then by (A) 5 Taking these logs. in Briggs' system, and reducing, we find. which is manifestly the true result; for in this case the gene and x is evi for, by the definition of a log. in the equation n=a3, x is the log. n. In like manner, n h log.(nh) h log. n. = n EXAMPLES FOR EXERCISE. Ex. 1. Given 22 +2′ =12 to find the value of x. y. Ans. x={a+log. n÷log.m} and y={a-log. n÷log. m}. Ex. 3. Given m* n* =a, and hx=ky to find x and y. x=log. a÷(log. m+log. n) and y=log. a÷(log. m+log. n). k To find the logarithm of any given number. Let N be any given number whose logarithm is tem whose base is a; then a=N and ax2=N2; sys hence, by the exponential theorem, we have from the last equation 1 1.2 |