What methods can I use to show that $2^{50} < 3^{33}$, without a calculator How would I show that $2^{50} < 3^{33}$, without a calculator, and what different methods are there of doing this?
Any help would be much appreciated. 
Thanks.
P.S sorry if the tag on this post is wrong. I wasn't sure what to put.

Method 1: Logarithms
Log both sides (since log is increasing) to get a STP:
$50\log 2<33\log 3$
Then compute with a calculator and solve.

Method 2: Approximation (of logarithms)
To make things simpler, we use log base 2. Then we want to show:
$50<33\log_23$.
Since $2^\frac{1}{2}\approx1.414$, we have $\log_23>>\frac{3}{2}$ (>> means by a decent amount). But then $33\log_23>>49.5$, so it is pretty close. I guess for maximum precision you'd need to use better approximations. (At this point I honestly cannot see how to solve the question without a ton of brute-force calculations)
