Can a Taylor series approximation be used to solve a mixed logarithmic equation?

[Esd]

"mixed" logarithmic equation

Homework Statement

I'm trying to solve for x in the following equation: e^(-66/x)/x^2 = c, where c is a constant

Homework Equations

The Attempt at a Solution

By taking ln of both sides and then dividing by 2, I get to:

-33/x -ln(x) = c/2

Then, in order to get an approximate answer, I tried substituting e with its approximation up to a few decimal places, in the original problem statement, but that didn't get me far. I'm considering doing a Taylor series approximation for the 'ln(x)' term (i.e (x-1) - (x-1)^2 + (x-1)^3)) and then solving for x. Is there an easier and/or more precise and/or more direct way? Note, I understand that the taylor approximation I used is meant for x, such that |x| is at most 1. This is why I only went with 3 terms.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[QuarkCharmer]

Have you considered arranging it in such a way where the X could be factored out?

 

[Esd]

Considered? Yes. But can't think of how.

 

[Mentallic]

The equation can't be solved in terms of elementary functions - in other words, you can't factor out an x - so you will need to find an approximate solution to the problem.