Algebraic of a variable to itself

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The equation a/b = (x/c)^x presents a challenge due to x appearing in both the base and the exponent. Attempts to solve it using logarithmic methods lead to x*log(x/c) = log(a/b), which is unhelpful. The consensus is that this equation cannot be solved algebraically. Instead, the Lambert W function is suggested as a potential solution, as it serves as the inverse of the function xex. Utilizing the Lambert W function may provide a pathway to finding a solution.
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I have been trying to solve algebraicly an equation of the following form for x with little success because x appears in the base and the exponent. Any help would be greatly appreciated.

a/b=(x/c)^x

A log approach only yields x*log(x/c)=log (a/b) which really doesn't get me anywhere.
 
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Well, you won't go very far! This is impossible to solve algebraically.
 
Should be able to use the Lambert W function- it is defined as the inverse to the function xex.
 

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