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let be the function $y=ax\log(bx)$, here $a$ and $b$ are constants and $\log$ is the natural logarithm

how can i evaluate the inverse function of this in terms of the Lambert $W$-function ??

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2 Answers 2

up vote 5 down vote accepted

opt has given the Wolfram Alpha output. Here I give an explicit derivation for reference.

We rearrange like so:


multiply both sides by $b$:


and make things a little complicated:


We are now in a position to leverage the Lambert function:


and now solving for $x$ is a snap:


Some people are tempted to "simplify" $\exp(W(z))$ to $z/W(z)$ (as was done in the Wolfram Alpha output), but I don't recommend this, as it introduces an unneeded removable discontinuity when $z=0$ for the principal branch...

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