TITLE: Find $x$ from $3^x\cdot x^3 = 1$
QUESTION [16 upvotes]: I saw a question on internet, tried to solve but I can't:
\begin{equation}
  3^x\cdot x^3 = 1
\end{equation}
I get $\ln$ function and made some equalization and I reached that:
\begin{equation}
  -x\cdot\ln(3) = 3\cdot\ln(x)
\end{equation}
I could not continue after that. How does it solve?
Wolfram|Alpha says nearly $x\approx 0.757$.
I searched on Google and on that site but I could not found anything (I guess about the mathematical sign).

REPLY [13 votes]: Equations like this can be solved in terms of the Lambert W function, which is the inverse of $f(z)=ze^z$ (and generally they don't have closed-form solutions in more elementary functions).  Specifically, consider the equation $y \ln y = e^{\ln y}\ln y = k$.  This inverts to $\ln y=W(k),$ or $y=e^{W(k)}$.  In your case, you have
$$
\frac{\ln 3}{3}=-\frac{1}{x}\ln x=\frac{1}{x}\ln\frac{1}{x},
$$
implying that
$$
\frac{1}{x}=\exp\left[W\left(\frac{\ln 3}{3}\right)\right]=e^{W(\ln \sqrt[3]{3})},
$$
or
$$
x=e^{-W(\ln \sqrt[3]{3})}.
$$
This correctly evaluates to $0.7576969788...$