$2^{50} < 3^{32}$ using elementary number theory How would you prove; without big calculations that involve calculator, program or log table; or calculus that

$2^{50} < 3^{32}$ 

using elementary number theory only?
If it helps you: $2^{50} - 3^{32} = -727120282009217$, $3^{32} \approx$ $2^{50.718800023077\ldots}$, $3^{32} $ $\div 2^{50}$ $=$ $1.6458125430068558$ (thanks to Henry).

Compare:
$$3^{32}=(3^{2})^{16}\quad\text{vs.}\quad2^{50}=4(2^{3})^{16}$$
So that using the binomial theorem to second order:
$$\frac{3^{32}}{2^{50}}=\frac{(9/8)^{16}}{4}=\frac{(1+1/8)^{16}}{4}
>\frac{1+16/8+120/64}{4}>1$$
