How many points in the xy-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect? The question in the title is equivalent to find the number of the zeros of the function $$f(x)=x^{12}-2^x$$
Geometrically, it is not hard to determine that there is one intersect in the second quadrant. And when $x>0$, $x^{12}=2^x$ is equivalent to $\log x=\frac{\log 2}{12}x$. There are two intersects since $\frac{\log 2}{12}<e$. 

Is there other quicker way to show
  this?


Edit: This question is motivated from a GRE math subject test problem which is a multiple-choice one(A. None B. One C. Two D.Three E. Four). Usually, the ability for a student to solve such problem as quickly as possible may be valuable at least for this kind of test. In this particular case, geometric intuition may be misleading if one simply sketch the curve of two functions to find the possible intersect.     

From Maple...
$$
\Biggl[\frac{-12\mathrm{LambertW} \biggl(\frac{\operatorname{ln} (2)}{12}\biggr)}{\operatorname{ln} (2)},\frac{-12\mathrm{LambertW} \biggl(\frac{-\operatorname{ln} (2)}{12}\biggr)}{\operatorname{ln} (2)},
\frac{-12\mathrm{LambertW} \biggl(-1,\frac{-\operatorname{ln} (2)}{12}\biggr)}{\operatorname{ln} (2)}\Biggr]
$$
$=[-0.9467803304,1.063346831,74.66932553]$ plus many complex solutions.
