ABSTRACT

In this paper, we study the equation of the form of $y^{\frac{x}{y}}=x$ which can also be written as $\frac{\ln x}{x}=\frac{\ln y}{y}$ . Apart from the trivial solution $x=y$ , a non-trivial solution can be expressed in terms of Lambert W function as $y=\frac{W\left[-\frac{\ln \left(x\right)}{x}\right]}{\left[-\frac{\ln \left(x\right)}{x}\right]}$ . For $y>\text{e}$ , 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 $y^{x/y^2}=x/y$ and $y^{x/y^3}=x/y^2$ written in base $y$ are exactly one and two orders higher respectively than the solutions of the equation $y^{\frac{x}{y}}=x$ . 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%.

Keywords:

Exponential Equation, Lambert W Function, Fine Structure Constant

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

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

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

$\frac{\ln x}{x}=\frac{\ln y}{y}$ (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 $\frac{\ln x}{x}$ . The plot indicates that, for any value of the function $\frac{\ln x}{x}$ 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 $\frac{\ln x}{x}=1/\text{e}=0.3679$ (rounded). For any value of y between e and infinity, a solution for x can be found in-between 1 and e.

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

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

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

Equation (6) can be written as

$\frac{\ln x}{\ln y}=\frac{x}{y^3}+2$ (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

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

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

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

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, $\ln \left(1.3713\right)/1.3713=0.2302$ 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.

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 ${\alpha }^{-1}$ .

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

References