Back
 JAMP  Vol.5 No.2 , February 2017
Solutions of the Exponential Equation yx/y = x or lnx/x = lny/y and Fine Structure Constant
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%.

1. Introduction

Lambert W function is a transcendental function [1] [2] which has applications in many areas of science which include QCD renormalisation, Planck’s spectral distribution law, water movement in soil and population growth [3] - [8] .

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.

Figure 1. The plot x vs ln ( x ) / x .

The solution of Equations ((1)-(3)) can be written in terms of Lambert W function [9] ,

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

If x = e , y = W [ 1 e ] [ 1 e ] and according to Dence [2] , 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.

Table 1. 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.

Figure 2. Solutions of x in base 10 and in base y for Equation (1) for y values of 1 to 13.

Figure 3. Solutions of x in base 10 and in base y for Equation (5) for y values of 1 to 13.

Figure 4. Solutions of x in base 10 and in base y for Equation (6) for y values of 1 to 13.

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 [10] 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 [11] . There could be a connection between Equations ((1) to (8)) and α 1 .

Cite this paper: Gnanarajan, S. (2017) Solutions of the Exponential Equation yx/y = x or lnx/x = lny/y and Fine Structure Constant. Journal of Applied Mathematics and Physics, 5, 386-391. doi: 10.4236/jamp.2017.52034.
References

[1]   Dence, T.P. (2013) A Brief Look into the Lambert W Function. Applied Mathematics, 4, 887.
https://doi.org/10.4236/am.2013.46122

[2]   Kalman, D. (2001) A Generalized Logarithm for Exponential-Linear Equations. College Mathematics Journal, 32, 2-14.
https://doi.org/10.2307/2687213

[3]   Corless, R.M., Gonnet, G.H., Hare, D.E., Jeffrey, D.J. and Knuth, D.E. (1996) On the Lambert W Function. Advances in Computational Mathematics, 5, 329-359.
https://doi.org/10.1007/BF02124750

[4]   Barry, D.A., et al. (2000) Analytical Approximations for Real Values of the Lambert W-Function. Mathematics and Computers in Simulation, 53, 95-103.
https://doi.org/10.1016/S0378-4754(00)00172-5

[5]   Valluri, S.R., Jeffrey, D.J. and Corless, R.M. (2000) Some Applications of the Lambert W Function to Physics. Canadian Journal of Physics, 78, 823-831.

[6]   Barry, D.A., et al. (1993) A Class of Exact Solutions for Richards’ Equation. Journal of Hydrology, 142, 29-46.
https://doi.org/10.1016/0022-1694(93)90003-R

[7]   Jain, A. and Kapoor, A. (2004) Exact Analytical Solutions of the Parameters of Real Solar Cells Using Lambert W-Function. Solar Energy Materials and Solar Cells, 81, 269-277.
https://doi.org/10.1016/j.solmat.2003.11.018

[8]   Scott, T.C., Mann, R. and Martinez II, R.E. (2006) General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function. A Generalization of the Lambert W Function. Applicable Algebra in Engineering, Communication and Computing, 17, 41-47.
https://doi.org/10.1007/s00200-006-0196-1

[9]   Weisstein, E.W. (2016) Lambert W-Function. From MathWorld—A Wolfram Web Resource.
http://mathworld.wolfram.com/LambertW-Function.html

[10]   Allen, H.S. (1915) Numerical Relations between Electronic and Atomic Constants. Proceedings of the Physical Society (London), 27, 425-431.
https://doi.org/10.1088/1478-7814/27/1/331

[11]   Kragh, H. (2003) Magic Number: A Partial History of the Fine-Structure Constant. Archive for History of Exact Sciences, 57, 395-431.

 
 
Top