TITLE: Comparing Powers of Different Bases
QUESTION [10 upvotes]: How can I know if one power is bigger than the other when the bases are different?
For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? I'll be working with big numbers, and though a more general solution is really appreciated, I will be comparing exactly powers of $2$ and $10$.

REPLY [3 votes]: Actually, you can solve problem above by much much easier method. No need for logarithm. 
Simply, give 2^2000 and 10^800 the same exponent. We know that (a^n)^m = a^(n*m). That is clear.
Now, we can restate the different bases but with the same exponent so they'll look like this: (2^5)^400 and (10^2)^400 and they are absolutely the same as the before ones. 2^5 = 32 and 10^2 = 100 are easily calculated, even without calculator, so now we can say that 32^400 < 100^400.
Nice and easy, without any trouble and logarithms :)
Hope that helped