How does one go about solving $x = \log_2{x}$? Is there a technique to solve these sorts of problems?
How to solve problems such as $x = \log_2{x}$
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logarithms
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1Use the method of Newton - Raphson to find a root of the equation $x-\log_2x=0$. See http://en.wikipedia.org/wiki/Newton%27s_method – 2012-12-04
3 Answers
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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 $(1,\infty)$.