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This question already has an answer here:

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. Thanks !!!

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marked as duplicate by Aaron Maroja, Jeremy Rickard, qwr, user147263, colormegone Feb 6 '16 at 0:10

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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The trick here is the Lambert W function, an inverse of $we^w$. We have $$x=\sqrt{2}^x=e^{x\ln\sqrt{2}},\,-x\ln\sqrt{2}e^{-x\ln\sqrt{2}}=-\ln\sqrt{2},\,x=-\frac{-W\left(-\ln\sqrt{2}\right)}{\ln\sqrt{2}}.$$ That's the theory, anyway. Like complex exponentiation, the Lambert W-function has multiple branches in $\mathbb{C}$. But the Lambert W function is of interest because many equations' solutions are expressible in terms of it.

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