If $x^x = 3$, how can I find x? I know I can rearrange this to $\log_x(3) = x$, and that some calculators can solve this, but how would you do this manually?
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2Do you know about the Lambert W function? Do you know how to get $x = e^{W(\log(3))}$ – Bumblebee Apr 28 '23 at 00:41
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2Does this answer your question? https://math.stackexchange.com/questions/532336/how-to-solve-equations-of-this-form-xx-n – Scene Apr 28 '23 at 00:42
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1Subtract the equation by $3$ to obtain $x^x-3 = 0$ and then use Newton's Method to approximate the root. – RyRy the Fly Guy Apr 28 '23 at 00:47
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Draw graph of $ y= x^x $ and $ y=3 $ the intersection point is the solution for the given equation. – Nitesh Apr 28 '23 at 00:55
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First, we have $x \ln x = \ln 3$. Here, $\ln x$ represents the natural logarithm. So $$e^{\ln }\ln = \ln 3.$$ Therefore, \begin{align} \ln x &= W( \ln 3 ) \\\implies x &= e^{W( \ln 3)} \end{align}
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I looked up the Lambert W function, but I don't really get how this helps. What is W? – Phenomniverse Apr 28 '23 at 01:27
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