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Trying to solve x*(x^x)=a for x, this is what I tried:

$$x*(x^x)=a$$

$$x^{x+1}=a$$

$$\ln(x^{x+1})=\ln(a)$$

$$(x+1)\ln(x)=\ln(a)$$

$$x+1=\frac{\ln(a)}{\ln(x)}$$

$$x=-1+\dfrac{\ln a}{\ln x}$$

But this still has x on the other side, what is x for this equation?

Hi John,

I don't think this expression has a closed form formula for x. The best you can hope for is a series or numerical solution to get an approximation. I know this isn't what you are after but I don't know how else to do the inversion.

Nonetheless, some progress can be made by manipulating the expression a little. Like you have done, taking the natural log is helpful:

x^(x+1) = a --> xln(x) + ln(x) = ln(a) --> xln(x) = c - ln(x) where c = ln(a)

If we plot the left and right hand sides of the last equation vs.x, the intersection of the curves gives the value of x the a given constant c. I've plotted these curves in the attached image. I've also plotted the function xln(x) + ln(x). Note that the intersection of this curve and the value of the constant c (on the y axis) corresponds to the same value of x as shown by the intersection of the other 2 curves.

Hope this helps.

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