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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?

pooja somani
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Sid
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  • See this question. – Dietrich Burde Oct 21 '18 at 11:42
  • It's a transcendental equation. You can not solve it in elementary functions. – Michael Rozenberg Oct 21 '18 at 11:44
  • Depends on what kind of solutions you are looking for. If you want integer solutions, then arithmetic is you friend :-) – Nicolas FRANCOIS Oct 21 '18 at 11:46
  • @NicolasFRANCOIS: a grapher quickly shows that there are no integer solutions, and you can formalize by exhibiting integers between which the curves cross.. –  Oct 21 '18 at 11:52
  • @DietrichBurde: IMO a very different question. The one is asking "why?", the other "how?". –  Oct 21 '18 at 11:58
  • @YvesDaoust You are right. It should be a duplicate of the first one. This type of question is also popular, I think. – Dietrich Burde Oct 21 '18 at 11:59
  • @YvesDaoust : yeah, but with arithmetic arguments, you don't need a grapher : $x$ must be a power of $2$, $x=2^k$, which leads to $2^{2^k}=2^{8k}$, $2^{k-3}=k$, and you stay with only a few cases to study, all of which failing (or you can continue saying $k$ must be a power of $2$...) – Nicolas FRANCOIS Oct 21 '18 at 12:02
  • @Yves Daoust, you are right, it's not really a match, but there is an answer there explaining "how". – Ennar Oct 21 '18 at 12:05

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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.

Claude Leibovici
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