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L^XVIII. Olserval'iom on the Solution of Exponential Equd- 

 iio/is. By G. A. Walker Arnott, Esq. Edinburgh. 



To Mr. TiUochi 



Sir, — XJ.AVING lately had occasion to enter upon some alge- 

 braic piobleuis involving exponential equations, I was very much 

 surprised at the manner of solution used by most writers on the 

 subject; who, instead of finding a direct approximation for the 

 purpose, had recourse to what is commonly called The Rule of 

 Trial and Error, or Double Position. 'I'his may be preferable in 

 some, nay in many cases^ and then it gives by far the quickest 



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mode ; but in others, as in the equation x =a, it gives, after one 

 or two trials^ an answer by no means so accurate as might be 

 desired. This I ascribe to the confused mixture of parts of 

 logarithms with the numbers themselves. 



Thinking, from the mode of solution universally employed, 

 that there migiit be none other known, or at least none so sim- 

 ple, as that its directness of approximation might compensate 

 for the difficulty of solution, I considered that some general for- 

 mula might easily be discovered, by attending to the nature of' 

 logarithmic series; and I was not disappointed in my researches 

 on this point. It is to the explanation of some of these formulae 

 that I intend to devote the following pages. 



L'nder exponential equations, we coitiprehehd those which have 

 for the index of one of the sides a variable or unknown quantity. 

 They may all be divided into three classes. Under the first come 

 those defined by the formida l^ = a, or where the exponent of 

 the power only is variable. Under the second class come those 



defined bv the formula x^ = o, or — - = a, which is the same with 

 x^ — — ; in these both the quantity and its index are vai'iable 



and equal. The third and last class comeS under the formula 

 jr = a^, where both the quantity and .the exponent of its root are 

 variable and equal. Besides these many more might be enu- 

 merated, such as x^=o, the general formula of our second class; 



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also a modification of this, x^ = a, or x^=n'^'^ the equation to 

 what is called the exponential curvcj which is in fact a logarith- 

 mic curve, whose subtangent or modulus is ttj— > and nianymore. 



But as these are indeterminate equations, unless y be some 

 function of x; and as it is only those capable of one direct an- 

 swer that I am about to examine, I do not think it necessary 

 Vol. 49. No. 229. May 1S17. X to 



