Is it possible to solve for $t$ in $t=x^t$? Using log on both sides does not seem to help. $$\log t=t\log x$$ $$\log x=\frac{\log t}t$$
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in base 10. If it is natural, what would be the difference? – John Glenn Mar 07 '18 at 14:57
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1If you are asking whether you can write $t=f(x)$ where the function $f$ is composed of "elementary" functions, the answer is no. And it does not matter what the base of the logarithm is. – NickD Mar 07 '18 at 14:59
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$(t,x)=(1,1)$ or $(-1,-1)$.......integer solutions – Of course it's not me Mar 07 '18 at 15:04
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This is done using Lambert's W function. The solution for this particular equation is
$$t = \frac{W\left(-\log x\right)}{-\log x} = e^{-W(-\log x)}=h(x),$$
where $h$ is Euler's iterated exponential.
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