of Exponential Equations. 323 



ting instead of the exponent x, its value a% and always repeating 



(J.\ &c. 



the operation, we have at length x = a^ 



.1 will hv n.) tneaiis sav that these series are easily api)licable 

 to practice: far from it, it is almost impossible to use them. We 

 mav there reject tliem as inconvenient; and I shall point out an- 

 other, which is not attended with such difficulty in the applica- 

 tion. 



Let x^ = a, then xLx = La', and \et x=b + y, where b is the 



nearest integer or approximate value of a;. Hence j; = ^M + jjf 



ahd Lx = L ('b{\ 4- 'i )) = hb + Lr+~| = LZ- + M x 



Ct" "*^^^ + Jzi" ~ ^*^') ^y ^^^^ nature of logarithmic series, 

 M being the modulus ; or, omitting all but the first two terms, 

 as it is only an approximation we desire, and multiplying by 



x—b + 7/, we have xLx=La={b-\-y)(Lb+ -^ ^^ — 



ILb + yLb+My J- + -^-^ omitting any term that in- 

 volves the cube of y; by arranging and transposing there- 

 fore, we have tjt y'" + y{M + Lb) =:La — bLb; and Jience 



y^ + 2y(b + ^-') = 2i(^ - "^y, or, what is simpler, y^ + 



2y{b + bHLb)z= 2b {ULa — bHLb), which quadratic equation 

 being solved, and added to b, gives 



x=b-{-y = J iQ^ +^+ ^')- ^', or simpler thus, 



a;=Ay^(2HLa + i + ^'HD^Z'J-^'I-lU. 



Example. Leta:*=100, and therefore Z' = 3. Then if we 

 use hyperbolic logarithms, 



x= \/ 27-63102 + 9+ 10vS625(j'-3-29584=6-8915 -3-2958 = 

 3*5957, or 8-59 + . Assuming b = o-b9, we may get a second 

 approximation, and then 



X = x/3;MJ65i220 +T2"888l + 2R)549639'-4-588565 = 

 8- 185853— 4 -588565 = 3-597288, which is true to the last fi- 

 gure, which should be 5. 



As, however, this mode requires the extraction of the square 

 root, and other intricate operations, the following is preferable, 

 as it docs not re(|uire some of these, and is after the first ap- 

 proximation equally correct with the other. Let us to derive 



X 2 this 



