inverse function of $y=ax\ln(bx)$

Question

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 ??

Answer 1

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

We rearrange like so:

$$\frac{y}{a}=x\log\,bx$$

multiply both sides by $b$:

$$\frac{by}{a}=bx\log\,bx$$

and make things a little complicated:

$$\frac{by}{a}=\exp(\log\,bx)\log\,bx$$

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

$$\log\,bx=W\left(\frac{by}{a}\right)$$

and now solving for $x$ is a snap:

$$x=\frac1{b}\exp\left(W\left(\frac{by}{a}\right)\right)$$

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...

Answer 2

inverse function of a x log(b x)