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
The Equation (1) can be written as
Converting the Equation (2) in terms of natural log gives
Equations ((1)-(3)) have a trivial solution , but they also have a non- trivial solution.
Figure 1 shows the plot of the function . The plot indicates that, for any value of the function in the range of 1 to infinity, it has two different solutions of . i.e. for any value of between 1 and infinity, a non-trivial solution of can be found. The plot also indicates that, at , there is only one solution and (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 .
The solution of Equations ((1)-(3)) can be written in terms of Lambert W function  ,
If , and according to Dence  , , hence , which is the result obtained graphically and numerically.
Some variations of Equation (1) are:
Equation (5) can be written as
Equation (6) can be written as
Equation (5) and Equation (6) have trivial solutions of and respectively.
2. Non-Trivial Solutions
If then Equation (1) becomes and (rounded) is the nontrivial solution, i.e. 100.13713 = 1.3713 and
If then Equation (5) and Equation (6) become and 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 and , Equation (4) gives
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.
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 , 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, and , 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 for any value.
The non-trivial solution for of Equation (6), 137.128857 is within 0.1% of the reciprocal value of the atomic fine structure constant , 137.0359991.
4. Possible Connection to Fine Structure Constant
Allen suggested that  however for the current values of and , the relationship is . Edward Teller suggested ln , where is the age of the universe  . There could be a connection between Equations ((1) to (8)) and .
 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.
 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.