How do you find an analytical solution for 3^xx=4?
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closed as off topic by J.C. Ottem, Denis Serre, quid, Daniel Litt, Will Jagy Sep 13 '11 at 21:21Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question. 


If you want, you can use Lambert's W function. Let $z:=x+4$, so that the equation becomes $3^{z4}=z$, so that $3^{4}=z3^{z}$, which upon inverting with Lambert's $W$ gives: $x=\frac{W(\ln(3)3^{4})}{\ln(3)}4$ In particular, $W(r)$ has a nice taylor expansion: $W(r)=\sum_{n=1}^\infty \frac{(1)^{n1}n^{n2}}{(n1)!}r^n$ 

