Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ??

share|improve this question

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:

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

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...

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.