TITLE:
Solutions of the Exponential Equation yx/y = x or lnx/x = lny/y and Fine Structure Constant
AUTHORS:
Sabaratnasingam Gnanarajan
KEYWORDS:
Exponential Equation, Lambert W Function, Fine Structure Constant
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.5 No.2,
February
17,
2017
ABSTRACT: In this paper, we study the equation of the form of which can also be written as . Apart from the trivial solution x = y, a non-trivial solution can be expressed in terms of Lambert W function as . For y > e, the solutions of x are in-between 1 and e. For integer y values between 4 and 12, the solutions of x written in base y are in-between 1.333 and 1.389. The non-trivial solutions of the equations and written in base y are exactly one and two orders higher respectively than the solutions of the equation . If y = 10, the rounded nontrivial solutions for the three equations are 1.3713, 13.713 and 137.13, i.e. 100.13713 = 1.3713. Further, ln(1.3713)/1.3713 = 0.2302 and W(-0.2302) = -2.302. The value 137.13 is very close to the fine structure constant value of 137.04 within 0.1%.