Simple product log

Algebra Level 3

Solve for $x$ : $\large W(x)=6$

Notation: $W(\cdot)$ denotes the product log which is the inverse function of $xe^x$

Please also have a look at this question about inverses of hyperoperations as well if you may have any solutions to that.

2 solutions

James Watson
Aug 17, 2020

$\begin{aligned} W(x) &= 6 \\ \implies x &= \green{\boxed{6e^6}} = \green{\boxed{2420.572761 \dots}} \end{aligned}$

Richard Desper
Aug 17, 2020

Cute.

$W(x)$ is the inverse function of $f(x) = x \exp(x)$ , so what value of $x$ satisfies? $W(x) = 6$ ?

Same question as what is $f(6)$ ? $W(x)$ is a red herring.