Find $x$ from $3^x\cdot x^3 = 1$ 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).

I don't believe the equation
$$-x\ln 3=3\ln x\iff 3\ln x+x\ln 3=0$$
can be solved by radicals, the solution probably no closed form . You can do the following however:
The function $$f(x)=3\ln x+x\ln 3$$ is strictly increasing and so it has at most one root. In addition, $f(\frac13)=-3\ln 3+\frac13\ln 3<0$ while $f(3)=6\ln 3>0$ and because $f$ is continuous, $f$ has a root in $(\frac13,3)$ (Intermediate Value Theorem). Therefore, there is only one solution of your equation and is in $(\frac13,3)$
