JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2017.52034JAMP-74192ArticlesPhysics&Mathematics Solutions of the Exponential Equation <i>y</i><sup><i>x/y</i></sup> = <i>x</i> or ln<i>x</i>/<i>x</i> = ln<i>y</i>/<i>y</i> and Fine Structure Constant SabaratnasingamGnanarajan1CSIRO Manufacturing, Lindfield, Australia* E-mail:15022017050238639129, November 201614, February 2017 17, February 2017© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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 . 10 0.13713 = 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%.

Exponential Equation Lambert W Function Fine Structure Constant
1. Introduction

Lambert W function is a transcendental function   which has applications in many areas of science which include QCD renormalisation, Planck’s spectral distribution law, water movement in soil and population growth  -  .

Considering the equation

y x y = x (1)

The Equation (1) can be written as

log y x = x y (2)

Converting the Equation (2) in terms of natural log gives

ln x x = ln y y (3)

Equations ((1)-(3)) have a trivial solution x = y , but they also have a non- trivial solution.

Figure 1 shows the plot of the function ln x x . The plot indicates that, for any value of the function ln x x in the range of 1 to infinity, it has two different solutions of x . i.e. for any value of y between 1 and infinity, a non-trivial solution of x can be found. The plot also indicates that, at y = e , there is only one solution x = e and ln x x = 1 / e = 0.3679 (rounded). For any value of y between e and infinity, a solution for x can be found in-between 1 and e.

The solution of Equations ((1)-(3)) can be written in terms of Lambert W function  ,

y = W [ − ln ( x ) x ] [ − ln ( x ) x ] (4)

If x = e , y = W [ − 1 e ] [ − 1 e ] and according to Dence  , W [ − 1 e ] = − 1 , hence y = e , which is the result obtained graphically and numerically.

Some variations of Equation (1) are:

y x / y 2 = x / y (5)

y x / y 3 = x / y 2 (6)

Equation (5) can be written as

ln x ln y = x y 2 + 1 (7)

Equation (6) can be written as

ln x ln y = x y 3 + 2 (8)

Equation (5) and Equation (6) have trivial solutions of x = y 2 and x = y 3 respectively.

2. Non-Trivial Solutions

If y = 10 then Equation (1) becomes 10 ( x / 10 ) = x and x = 1.3713 (rounded) is the nontrivial solution, i.e. 100.13713 = 1.3713 and

ln x x = ln y y = 0.2302

If y = 10 then Equation (5) and Equation (6) become 10 ( x / 100 ) = x / 10 and 10 ( x / 1000 ) = x / 100 respectively and their solutions are 13.713 (rounded) and 137.13 (rounded) respectively. These solutions are exactly one and two orders larger than the solution of Equation (1).

Also if x = 1.3713 and y = 10 , Equation (4) gives

W [ − ln ( 1.3713 ) 1.3713 ] = 10 [ − ln ( 1.3713 ) 1.3713 ]

Hence W ( − 0.2302 ) = − 2.302

For the range of integer y values of 4 to 12, the non-trivial solutions for x of Equations ((1), (5) and (6)) were obtained using iterative method. The solutions of x are written in base 10 and in base y (Table 1). Plots of y vs x with x in base 10 and in base y are shown in Figures 2-4 respectively.

3. Conclusions

The non-trivial solutions of Equations ((1), (5) and (6)) written in base y, differ exactly by one order. For y values in the range of 4 to 12, the solutions of Equation (6) written in base y are in the range of 133.33 to 138.99.

When y = 10 , the rounded nontrivial solutions for Equation (1), Equation (5) and Equation (6) are 1.3713, 13.713 and 137.13, i.e. 100.13713 = 1.3713, ln ( 1.3713 ) / 1.3713 = 0.2302 and W ( − 0.2302 ) = − 2.302 , i.e. for the argument values of 1.3713 and −0.2302, the function values are exactly one order higher. To our knowledge, these results were not reported before.

Rounded non-trivial solutions for x of Equations ((1), (5) and (6)) for y values from 4 to 12 are written in base 10 and base y
ySolutions of Equation (1)Solutions of Equation (5)Solutions of Equation (6)
In base 10In base yIn base 10In base yIn base 10In base y
121.31221.38915.7513.89189.0138.9
111.33891.38014.7313.80162.0138.0
101.37131.37113.7113.71137.1137.1
91.41141.36312.7013.63114.3136.3
81.46251.35511.7013.5593.6135.5
71.53011.35010.7113.5075.0135.0
61.62421.3439.7513.4258.5134.3
51.76491.3408.8213.4044.11340
42.00001.333..8.0013.33..32.0133.3

The trivial solutions of Equations ((1), (5) and (6)) can be written as 10, 100 and 1000 in base y for any y value.

The non-trivial solution for x of Equation (6), 137.128857 is within 0.1% of the reciprocal value of the atomic fine structure constant α − 1 , 137.0359991.

4. Possible Connection to Fine Structure Constant

Allen suggested that m e / M p ~ 10 α 2  however for the current values of m e / M p and α , the relationship is m e / M p = 10.227 α 2 . Edward Teller suggested ln T 0 3 / 2 = α − 1 , where T o is the age of the universe  . There could be a connection between Equations ((1) to (8)) and α − 1 .

Cite this paper

Gnanarajan, S. (2017) Solutions of the Exponential Equation or and Fine Structure Constant. Journal of Applied Mathematics and Physics, 5, 386-391. https://doi.org/10.4236/jamp.2017.52034

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