I was recently lecturing on the hierarchy of functions in a calculus class. Discussing the fact that eventually any exponential growth function will overcome any polynomial. As an example, I tried asking Mathematica the following:
NSolve[q^800 + 4000000 == (1000001/1000000)^q, q, Reals] and it completely choked.
Honestly, I'm only interested in the order of magnitude of the answer, so that I can say, "The exponential doesn't exceed the polynomial until you input ???-digit numbers, but eventually it does happen."
How can I get Mathematica to solve this just to an order of magnitude, i.e. how do I get Mathematica to just provide a really gross estimate?