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How does one go about solving $x = \log_2{x}$? Is there a technique to solve these sorts of problems?

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Use the method of Newton - Raphson to find a root of the equation $x-\log_2x=0$. See – MathOverview Dec 4 '12 at 12:27

There are no elementary formulas for such equations. Using the Lambert W function, the solution is $x = e^{-W(-\log2)}$.

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This can be "solved" using the Lambert W function. The example on the wikipedia page is fundamentally the same as your question, after rewriting as $2^x=x$.

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There are. Look at it as an intersection of two functions - $$f(x)=x$$ and $$g(x)=\log_2(x)=\ln(x)/\ln(2)$$ both functions are monotone increasing. Find their derivatives, and check if they can ever meet, and how many times. Separate the search to $(0,1)$ and $(1,\infty)$.

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