Solutions of the Exponential Equation yx/y = x or lnx/x = lny/y and Fine Structure Constant

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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

${y}^{\frac{x}{y}}=x$ (1)

The Equation (1) can be written as

${\mathrm{log}}_{y}x=\frac{x}{y}$ (2)

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

$\frac{\mathrm{ln}x}{x}=\frac{\mathrm{ln}y}{y}$ (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 $\frac{\mathrm{ln}x}{x}$ . The plot indicates that, for any value of the function $\frac{\mathrm{ln}x}{x}$ 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 $y=\text{e}$ , there is only one solution $x=\text{e}$ and $\frac{\mathrm{ln}x}{x}=1/\text{e}=0.3679$ (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 $\mathrm{ln}\left(x\right)/x$ .

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

$y=\frac{W\left[-\frac{\mathrm{ln}\left(x\right)}{x}\right]}{\left[-\frac{\mathrm{ln}\left(x\right)}{x}\right]}$ (4)

If $x=\text{e}$ , $y=\frac{W\left[-\frac{1}{\text{e}}\right]}{\left[-\frac{1}{\text{e}}\right]}$ and according to Dence [2] , $W\left[-\frac{1}{\text{e}}\right]=-1$ , hence $y=\text{e}$ , 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

$\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{{y}^{2}}+1$ (7)

Equation (6) can be written as

$\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{{y}^{3}}+2$ (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
${10}^{\left(x/10\right)}=x$ and
$x=1.3713$ (rounded) is the nontrivial solution, i.e. 10^{0.13713} = 1.3713 and

$\frac{\mathrm{ln}x}{x}=\frac{\mathrm{ln}y}{y}=0.2302$

If $y=10$ then Equation (5) and Equation (6) become ${10}^{\left(x/100\right)}=x/10$ and ${10}^{\left(x/1000\right)}=x/100$ 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

$W\left[-\frac{\mathrm{ln}\left(1.3713\right)}{1.3713}\right]=10\left[-\frac{\mathrm{ln}\left(1.3713\right)}{1.3713}\right]$

Hence $W\left(-0.2302\right)=-2.302$

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. 10^{0.13713} = 1.3713,
$\mathrm{ln}\left(1.3713\right)/1.3713=0.2302$ and
$W\left(-0.2302\right)=-2.302$ , 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 ${\alpha}^{-\text{1}}$ , 137.0359991.

4. Possible Connection to Fine Structure Constant

Allen suggested that ${m}_{e}/{M}_{p}~10{\alpha}^{2}$ [10] however for the current values of ${m}_{e}/{M}_{p}$ and $\alpha $ , the relationship is ${m}_{e}/{M}_{p}=10.227{\alpha}^{2}$ . Edward Teller suggested ln ${T}_{0}^{3/2}={\alpha}^{-1}$ , where ${T}_{o}$ is the age of the universe [11] . There could be a connection between Equations ((1) to (8)) and ${\alpha}^{-1}$ .

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.