solve $y=ax\ln(b/x)$ for $x$

Question

Is it possible to solve the above equation for $x$? Unfortunately WolframAlpha timed out without giving any hints.

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Answer 1

Using the Lambert W-function with the property $$e^{W(x)} = \frac{x}{W(x)}$$ then the following is obtained: \begin{align} y &= a \, x \, \ln\left(\frac{b}{x}\right) \\ - \frac{y}{a \, b} &= \frac{x}{b} \, \ln\left(\frac{x}{b}\right) = \ln\left(\frac{x}{b}\right) \, e^{\ln\left(\frac{x}{b}\right)} \\ W\left(- \frac{y}{a \, b}\right) &= \ln\left(\frac{x}{b}\right) \\ \frac{x}{b} &= e^{W(-y/(a b))} = - \frac{y}{a b \, W(-y/(a b))} \\ x &= - \frac{y}{a \, W\left(- \frac{y}{a \, b}\right)}. \end{align}

Answer 2

You can't solve it using elementary functions. You need to take a look at Lambert W Function