522 Observations on the Solution 



to comprehend them at all under my division. I may remark 

 besides, that exponential equations are divided into orders, ac- 

 cording to the number of exponents above one another ; thus 



X X 



x', a* are of the first order, x^, a', are of the second order ; and 

 30 on. This subdivision was, I believe, first noticed by Ber- 

 noulli. 



As to the first class, or those under the form l^ = a, I have 

 nothing new to say, the common mode of solution being quite 

 sufficient. I shall therefore point out how it may be obtained, 

 and take an example in illustration. Let then l'' = a; by taking 



the logarithms of both sides, we have xLb = La, or x= —' 



which gives the common rule. 



Example. Let 2-= 100. then x= ^= ?:^^22^ =6-64386. 



^ ■ L2 -30103 



This order of the first class is far simpler than any of the other 

 orders or classes, as we shall immediately find. As for the se- 



X 



cond order of this class, or l^ = a, that is x^hb^ha, or x^ = 

 — , it is evident that it comes under the firsi order of the fol- 

 lowing class. 



On account of the second class being as it were the key to the 

 others, I shall explain the rules of approximation rather more 

 minutely than those of the third. We shall also confine our 

 attention solely to the first and second orders, and shall explain 

 them separately. 



1st. Of those equations under the form x^=:a. These admit 

 of being converted into an infinite series, which, though not a 

 fast converging one, yet deserves a place here on account of its 

 simplicity. It may be derived as follows : 



Let x'^—a, by taking the logarithms, «Lx=La, or x= y— • 



Consequently Lx = L (■^ = '^L"a—'L"x', but instead of this 



last term, we may substitute the logarithm of the whole of 

 L"a — L"a;, then La; = L''a— L(L"a— L"x); and by similar sub- 

 stitutions continually repeated, we get Lj:=L"a — L(L"a — L) 

 (L"a— L) &c. ad iivfinitum. And x is equal to the correspond- 

 ing number. But x admits of being obtained by an infinite 



J. 

 scries without logarithms: thus x^rsa, or xssa", by substitu- 



* In tills and the following I make use of L" to denote the logarithm 

 of a lo^aritlmi, or the second logarithm ; L"' to denote the third logarithm : 

 and i>u ua. 



ting 



