Prove $17^{14}\gt 31^{11}$

Prove $17^{14}\gt 31^{11}$.

My solution:

$11\log_{17}31\lt11\log_{17}32\lt 11\log_{16}32=11\left(\dfrac{\log_{2}32}{\log_{2}16}\right)=11\left(\dfrac{5}{4}\right)=13.75\lt 14$

Therefore we get $17^{14}\gt 31^{11}$ and we're done!

See, if we can see the subtext of the problem and understand the situation thoroughly, we could deliver the proof easily and elegantly, within the shortest possible of time.:D

And that doesn't mean that is the only proof for the problem. Please share with me if you could solve it different than mine, and I look forward to hearing from you!