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I was able to solve this equation using graphical methods, but cannot figure out a mathematical solution to the equation.
What approach should be taken to solve it?

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  • $\begingroup$ See this question. $\endgroup$ – Dietrich Burde Oct 21 '18 at 11:42
  • $\begingroup$ It's a transcendental equation. You can not solve it in elementary functions. $\endgroup$ – Michael Rozenberg Oct 21 '18 at 11:44
  • $\begingroup$ Depends on what kind of solutions you are looking for. If you want integer solutions, then arithmetic is you friend :-) $\endgroup$ – Nicolas FRANCOIS Oct 21 '18 at 11:46
  • $\begingroup$ @NicolasFRANCOIS: a grapher quickly shows that there are no integer solutions, and you can formalize by exhibiting integers between which the curves cross.. $\endgroup$ – Yves Daoust Oct 21 '18 at 11:52
  • $\begingroup$ @DietrichBurde: IMO a very different question. The one is asking "why?", the other "how?". $\endgroup$ – Yves Daoust Oct 21 '18 at 11:58
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This equation has explicit solutions in terms of Lambert function. In the real domain, there are three roots $$x_{1,2}=-\frac{8 }{\log (2)}W_0\left(\pm\frac{\log (2)}{8}\right)$$ $$x_3=-\frac{8 }{\log (2)}W_{-1}\left(-\frac{\log (2)}{8}\right)$$ Have a look to the Wikipedia page for the manipulations and the series expansions for the numerical evaluation of them.

In the complex domain, I guess that there as much more.

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