Is there an algebraic method for finding roots involving logarithms?

[Telemachus]

Homework Statement

Perhaps this is trivial, but working in thermodynamics oftenly requires of handling with logarithms. For example, in an exercise I'm trying to solve, I need the roots for something like:
x+ln (x)=constant
The thing is I don't know how to find this kind of roots. I mean, I can use some approximation methods from calculus, like the Newton method for the roots approximation, but I thought that maybe there is other way of doing this, a more algebraic way. Does anyone know if there is some algebraic way of getting this kind off roots?

 

[I like Serena]

Homework Statement

Perhaps this is trivial, but working in thermodynamics oftenly requires of handling with logarithms. For example, in an exercise I'm trying to solve, I need the roots for something like:
x+ln (x)=constant
The thing is I don't know how to find this kind of roots. I mean, I can use some approximation methods from calculus, like the Newton method for the roots approximation, but I thought that maybe there is other way of doing this, a more algebraic way. Does anyone know if there is some algebraic way of getting this kind off roots?

There is a Lambert-W function, just for this problem (see wikipedia), which offers the "algebraic" solution to your problem.

The solution for your problem is then:

x=W(e^constant)

However, I'm afraid this doesn't help you much, since you would still need to calculate the W-function. So you remain stuck with a numerical approximation.

 

[Telemachus]

Thanks Serena, that's good enough for me.