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If we have the expression $a=x^{c\cdot x+1}$ where the values of $a,c$ are known, how can we find the value of $x$?

I tried using log but it yields: $x = a ^ {(1/x)/(c-1/x)}$ from which I can't find any solution.


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Solving this kind of equation usually requires the Lambert W function. – Arturo Magidin Oct 13 '11 at 17:14
...actually, after wrestling with this for a while, it doesn't look to me that this can be massaged into something where Lambert applies; you have an addition in the exponent, but none in the base. You may need to use Newton-Raphson for this... – J. M. Oct 14 '11 at 13:48
The straightforward iteration $x' = a^{1/(cx+1)}$ seems to converge (and quickly) for all positive $a,c$ (I have not proved it). – leonbloy Oct 14 '11 at 15:45

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