_{1}

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%.

Lambert W function is a transcendental function [

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.

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 [

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.

If y = 10 then Equation (1) becomes 10 ( x / 10 ) = x and x = 1.3713 (rounded) is the nontrivial solution, i.e. 10^{0.13713} = 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 (

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 ( 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.

y | Solutions of Equation (1) | Solutions of Equation (5) | Solutions of Equation (6) | |||
---|---|---|---|---|---|---|

In base 10 | In base y | In base 10 | In base y | In base 10 | In base y | |

12 | 1.3122 | 1.389 | 15.75 | 13.89 | 189.0 | 138.9 |

11 | 1.3389 | 1.380 | 14.73 | 13.80 | 162.0 | 138.0 |

10 | 1.3713 | 1.371 | 13.71 | 13.71 | 137.1 | 137.1 |

9 | 1.4114 | 1.363 | 12.70 | 13.63 | 114.3 | 136.3 |

8 | 1.4625 | 1.355 | 11.70 | 13.55 | 93.6 | 135.5 |

7 | 1.5301 | 1.350 | 10.71 | 13.50 | 75.0 | 135.0 |

6 | 1.6242 | 1.343 | 9.75 | 13.42 | 58.5 | 134.3 |

5 | 1.7649 | 1.340 | 8.82 | 13.40 | 44.1 | 1340 |

4 | 2.0000 | 1.333.. | 8.00 | 13.33.. | 32.0 | 133.3 |

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.

Allen suggested that m e / M p ~ 10 α 2 [

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