Q:

how to solve $\log{x}=cx^4$ for $x$

I was wondering if there is a general solution for this form of equations:
$$\log{x}=cx^4$$
Tried: $$ x = e^{cx^4}\\
xe^{-cx^4}=1$$

A:

Note that $\log x/x^4$ has derivative $\frac{1-4\log x}{x^5}$, so the maximum real value for $c$ is at $x=e^{1/4}$, which is a maximum value of $c=\frac{1}{4e}$.
This is one of those equations that requires the Lambert W-function.
Raise your last equation to the fourth power, then multiply by $-4c$ and you get:
$$-4cx^4e^{-4cx^4}=-4c$$
Then apply the Lambert W-function and you get:
$$-4cx^4=W(-4c)$$
Or $$x=\left(\frac{W(-4c)}{-4c}\right)^{1/4}$$
There are actually two real values of $W(-4c)$ when $0<c<\frac{1}{4e}$. See the linked article for details about this.

