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

You know that $3^7> 2000$ and $2^{10} = 10^3(1+r)$ with $|r| < 3/100 \ll 1$ hence
$$
2^{50}\simeq 10^{15} (1+5r)
\\
3^{32} = 3^4\times 3^{28} = 3^42^4 10^{12} > 1200 \times 10^{12
} > 2^{50}$$
