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Advanced Math/Solve this equation for x


Trying to solve x*(x^x)=a for x, this is what I tried:
$$x=-1+\dfrac{\ln a}{\ln x}$$

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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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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