&c. where the sign is put over the index, instead of before it, when that part of the logarithm is negative, in order to distinguish it from the decimal part, which is always to be considered as +, or affirmative. 518. Also, agreeably to what has been before observed, the logarithm of 38540 being 4.5859117, the logarithms of any other numbers, consisting of the same figures, will be as follows: Numbers. Logarithms. 3854 3.5859117 385.4 2.5859117 38.54 1.5859117 3.854 0.5859117 .3854 1.5859117 .03854 2.5859117 .003854 3.5859117 which logarithms, in this case, differ only in their indices, the decimal or positive part, being the same in them all. 519. And as the indices, or the integral parts of the logarithms of any numbers whatever, in this system, can always be thus readily found, from the simple consideration of the rule above-mentioned, they are generally omitted in the tables, being left to be supplied by the operator, as occasion requires. 520. It may here, also, be farther added, that, when the logarithm of a given number, in any particular system, is known, it will be easy to find the logarithm of the same number in any other system, by means of the equations, a*=n, and en, which give (1) x= log. n, x'= 1. n Where log. denotes the logarithm of n, in the system (2). of which a is the base, and 1. its logarithm in the system of which e is the base. 521. Whence a*=e* or a*=e, and ea, we shall х have, for the base a, log. e, and for the base e, = l.a; or (3) (4). . . x=x' log. e, x'=x.l.a Whence, if the values of x and x', in equations (1), (2), be substituted for x and x in equations (3), (4), х х 1 we shall have, log. n= log. exl.n, and l.n=. X 1 log.e log. n; or l.nl.a x log. n, and log. n= =l.n. 1 1.a where log. e, or its equal expresses the constant 1.a ratio which the logarithms of n have to each other in the systems to which they belong. 522. But the only system of these numbers, deserving of notice, except that above described, is the one that furnishes what have been usually called hyperbolic or Neperian logarithms, the base of which is 2.718281828459 523. Hence, in comparing this with the common or tabular logarithms, we shall have, by putting a in the latter of the above formulæ =10, the expression 1 log. n= xl.n, or l.n=1.10×log. n. 7.10 Where log., in this case, denotes the common logarithm of the number n, and 7. its Neperian logarithm; 4342944819. being what is usually called the modulus of the common or tabular system of logarithms. 524. It may not be improper to observe, that the logarithms of negative quantities, are imaginary; as has been clearly proved, by LACROIX, after the manner of EULER, in his Traité du Calcul Differentiel et Integral and also, by SUREMAIN-MISSERY in his Théorie Purement Algébrique des Quantités Imaginaires. See, for farther details upon the properties and calculation of logarithms, GARNIER's d'Algebre, or BONNYCASTLE'S Treatise on Algebra, in two vols. 8vo. II. APPLICATION OF LOGARITHMS TO THE SOLUTION 525. EXPONENTIAL EQUATIONS are such as contain quantities with unknown or variable indices: Thus, a*=b, x*=c, a*=d, &c. are exponential equations. Уу 526. An equation involving quantities of the form x, where the root and the index are both variable, or unknown, seldom occur in practice, we shall only point out the method of solving equations involving quantities of the form a*, ab, where the base a is constant or invariable. 527. It is proper to observe that an exponential of the form ab, means, a to the power of b3, and not ao to power of x. the Ex. 1. Find the value of x in the equation a*=b. Taking the logarithm of the equation ab, we have xx log. a=b; ·.x= log. b thus, let a=5, b= log. a Ex. 2. It is required to find the value of x in the equation a=c. Assume by, then a=c, and yx log. a= log. c; log.c Hence b log. (which let)=d. Take log. a the logarithm of the equation hd, then, by (Ex. 1.), Thus, let a=9, b=3, c=1000; then in log. a log. d log. b log. d_log. 3.14_.4969296 .4771213 = =1.04. Ex. 3. Make such a separation of the quantities in the equation (a2 —b2)*=a+b, as to show that Taking the logarithm, we have х xx log. (a3 —b2)= log. (a+b), or x × log. (a+b)× = log. (a-b) (a+b); that is, xx log. (a+b)+xx log. (a—b)= log. (a+b). Hence xx log. (a—b)=log. (a+b)—x × log. (a+b) log. (a+b) 1 xlog. (a—b)' Ex. 4. Given a+b2=c, and a*—by=d, required the values of x and y. By addition, 2a*=c+d, or a*: m; then x= 2 log. m log. a Again, by subtraction, we have 26"=c-d, or by— (which let =n) ; •'•y—log. b' Ex. 5. Find the value of x in the equation Ex. 6. Find the value of x in the equation ɑ*— 1 log. (b+c)+ log. (b—c)- log. d-4 log. e Ans. x= Ex. 8. Given log. x+ log. y= to find the vaand log.x-log. y=lues of x and y. Ans. x 10/10 and y= = =10. Ex. 9. In the equation 3*=10, it is required to find the value of x. Ans. x 3.32198, &c. Ex. 10. Given /729=3, required the value of xAns. x=6. Ex. 11. Given 57862-8, to find the value of x. Ans. 5.2734, &c. 3 Ex. 12. Given (216)=64, to find the value of 2. Ans. x=4.2098, &c. Ex. 13. Given 43-4096, to find the value of x. log. 3 Ex. 14. Given a+y=c, and byd, to find the va THE RESOLUTION OF EQUATIONS OF THE THIRD AND HIGHER DEGREES. I. THEORY AND TRANSFORMATION OF EQUATIONS: 528. In addition to what has been already said (Art. 192), it may here be observed, that the roots of any equation are the numbers, which, when substituted for the unknown quantity, will make both sides of the equation identically equal. Or, which is the same, the roots of any equation are the numbers, which, substituted for the unknown quantity, reduce |