science

[Not-so-daily puzzle] Which one is bigger?

south_korea_ln

$$
10^{36} > 9^{37}
$$

$$
10^{36} > 9 * 9^{36}
$$

$$
\frac{10^{36}}{9^{36}}  > 9
$$

$$
(\frac{10}{9})^{36}  > 9
$$

$$
(1 + \frac{1}{9})^{36}  > 9
$$

$$
((1 + \frac{1}{9})^9)^4 > 9
$$

Applying [Bernoulli's inequality](https://en.wikipedia.org/wiki/Bernoulli's_inequality) $$(1+x)^r \geq 1+rx$$ to the inner term

$$
(1+\frac{1}{9})^9 \geq 1+9*\frac{1}{9}
$$

$$
(1+\frac{1}{9})^9 \geq 2
$$

Inserting into the outer term

$$
2^4 > 9
$$
