GRE Subject question $\log x=c x^4$

Question

For what positive value of $c$ does the equation $\log x=cx^4$ have exactly one real solution for $x$?. Thank you.

Answer 1

Hint write it as $\frac {log (x)}{x^4} =c $ then notice values at $1,\infty $ they are $0$ value at infinity can be found out using limits.Thus using rolles theorem there is altleast $1 x $ for which $f'(d)=0$ for $d\in (0,\infty)$ thus you can find the $d $