Solving for x in k=(x)ln(x): A Daunting Task

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SUMMARY

The equation k=(x)ln(x) does not have a straightforward explicit solution for x. Attempts to manipulate the equation, such as rewriting it in exponential form or raising both sides to the power of -k, do not yield results. However, numerical methods can effectively solve the equation for specific values of k. It is important to recognize that while many equations in exercises have explicit solutions, this particular equation exemplifies cases where no such formula exists.

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Ja4Coltrane
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I was wondering how one would solve for x in the equation
k=(x)ln(x)

I tried all normal means. For example rewriting the equation with the x as an exponent of x. I have tried writing it in exponential form and the raising both sides to the power -k. I basically have no idea how to do this.
I do notice that the x sort of functions in the way that it would in a damping function...does that have anything to do with it?
 
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Many "simple looking" equations, including your equation, do not have solutions that can be written as an explicit formula "x = something".

It is easy to solve the equation numerically, for any given value of k.

Of course the equations that occur in exercises all DO have solutions, and that might mislead you into thinking that any equation can be solved explicitly.
 

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