How many or two many?

A common mistake is to think there are only one or two solutions (x=2,4). However we see there is a third solution by sketching the graph y = $x^4$ - $4^{x}$

Refer to the above image

At x = 0, $x^4$ = 0 and $4^{x}$ = 1 $\Rightarrow {x}^{4}<{4}^{x}$ .

At x = -1, $x^4$ = 1 and $4^{x}$ = $\frac{1}{4}$ $\Rightarrow {x}^{4}>{4}^{x}$ .

As both graphs y = $x^4$ and y = $4^{x}$ are continuous, they must cross over each other in the interval (-1,0) and we find the third and final solution there.