Is there an explicit way to write the solution of nx = e^x?

[Hiero]

(Not homework, just curious)

Homework Statement

Is there an explicit way to write the x which solves nx = e^x for constant n?

Homework Equations

nx = e^x
Or equivalently,
x = ln(x) + m

The Attempt at a Solution

For any n we could just find x numerically, but is there an explicit expression for x(n)?

 

[Alloymouse]

Been poking around the internet, but as far as I can see this has not been done :/

 

[phyzguy]

https://en.wikipedia.org/wiki/Lambert_W_function

 

[Ray Vickson]

(Not homework, just curious)

Homework Statement

Is there an explicit way to write the x which solves nx = e^x for constant n?

Homework Equations

nx = e^x
Or equivalently,
x = ln(x) + m

The Attempt at a Solution

For any n we could just find x numerically, but is there an explicit expression for x(n)?

It depends on what you mean by "explicit". Mathematicians have invented and named a function designed to deal with problems of that type; it is called the Lambert W-function, $W$. In terms of $W$ the solution is $x = -W(-1/n)$. The function $W$ is non-elementary, but has well-developed algorithms and procedures available for it---perhaps not on hand-held calculators or standard spreadsheets, but certainly in computer algebra systems. If you have such a system available to you, computing $W$ is no more exotic than computing a sine or a cosine---just newer and less familiar.