For what positive value of $c$ does the equation $\log(x)=cx^4$ have exactly one real root? For what positive value of $c$ does the equation $\log(x)=cx^4$ have exactly 
one real root?
I think I should find a way to apply IMV and Rolle's theorem to $f(x) = \log(x) - cx^4$. I think I should first find a range of values for $c$ such that the given equation has a solution and then try to find one that gives only a unique solution. I have thought over it but nothing comes to my mind. Perhaps I'm over-complicating it.
Any ideas on how to proceed are appreciated. 

One may use a slightly geometric argument for this problem. Notice that $\ln(x)=cx^4$ when $f(x)=cx^4-\ln x=0$. Now, $f$ has a vertex, and the vertex occurs where $f'(x)=4cx^3-\frac{1}{x}=0$. Thus, we have the system of equations
\begin{align}
f(x)&=cx^4-\ln x=0\\
f'(x)&=4cx^3-\frac{1}{x}=0.
\end{align}
Solve the second equation for $x^4$ to get $x^4=\frac{1}{4c}$. Substitute this into the first equation to get $\ln x = \frac{1}{4}$, which implies $x=e^{1/4}$. Finally, substitute this equation for $x$ into $\frac{1}{4}=\ln x=cx^4$ to find that $c=\frac{1}{4e}$.
