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%.