NOTE 4. The same rule also, among other more diffi cult forms of equations, succeeds very well in what are called exponential equations, or those, which have an unknown quantity for the exponent of the power; as in the following example. 4. To find the value of x in the exponential equation *100. For the more easy resolution of this kind of equations, it is convenient to take the logarithms of them, and then compute the terms by means of a table of logarithms. Thus, the logarithms of the two sides of the present equation are, xX log. of x=2, the log. of 100. Then by a few trials it is soon perceived, that the value of x is somewhere between the two numbers 3 and 4, and indeed nearly in the middle between them, but rather nearer the latter than the former. By taking therefore first x=3.5, and then x=3.6, and working with the logarithms, the operation will be as follows : As '098451: 'I :: '002689: 000273 Which correction, taken from 3.60000 Leaves 3.59727=x nearly. On trial, this is found to be very little too small. Take therefore again x=3.59727, and next x=3.59728, and repeat the operation as follows : 0.0000099 difference of the errors. Then, As 0000099: 00001 :: 0000047: 000000474747 Which correction, added to 3.597280000০০ Gives nearly the value of x = 3.59728474747 5. To find the value of x in the equation x3 +ICX*+ 5x=2600. Ans. x=1100673. 6. To find the value of x in the equation x3-2x5. 7. To find the value of x in the equation x3 +2x223x70. Ans. x=51349. 8. To find the value of x in the equation x3-17x2+ 2 2 54x350. Ans. 14.95407. 9. To find the value of x in the equation x-3x2Ans. x=10.2615. 75x=10000. 10. To find the value of x in the equation 2x4-16x3 2 +40x2-30x=-1. Ans. x=1284724. 11. To find the value of x in the equation x5 +2x++ 3x3+4x+5x=54321. Ans. x=8.414455. 12. To find the value of x in the equation x*= 123456789. Ans. x86400268. END OF ALGEBRA. GEOMETRY. DEFINITIONS, 1. A POINT is that, which has position, but not magnitude. 2. A line is length, without breadth or thickness. 3. A surface, or superficies, is an extension, or a figure, of two dimensions, length and breadth, but without thickness. 4. A body, or solid, is a figure of three dimensions, namely, length, breadth and thickness. Hence surfaces are the extremities of solids; lines the extremities of surfaces; and points the extremities of lines. 5. Lines * A TUTOR teaches Simson's Edition of EUCLID'S ELE TENTS OF GEOMETRY in Harvard College. |