The math textbook of a student that I'm working with posed the following problem:
The student told me that she couldn't do the problem and so, with an air of authority, I started to show her how to solve it...until I realized that I couldn't solve it, either! I believe that there is no closed-form analytical solution to such an equation, and that the best that one can do is to solve it with a graphing calculator, looking for the intersection(s) of f(x)=e^x and g(x)=x^5.
Another way to solve such an equation is with a spreadsheet, evaluating both functions at a sampling of x values, calculating the difference of those functional values, and modifying the x domain until the difference becomes "small". How small is small would have to be defined.