Which is bigger, $2^{20}$ or $20^{2}$ ?

Q: Which is bigger, 2^20 or 20^2?

2^20 is bigger: 2^20 = 1048576 and 20^2 = 400.

$2^{20}$ is bigger: $2^{20}$ = 1048576 and $20^{2}$ = 400.

2^{20} = 1048576\ >\ 20^{2} = 400

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^20 against 20^2 is the same question as which of the two marked points is higher; the curve peaks at x = e.

How to work it out

1
Compute both exactly: $2^{20}$ = 1048576 and $20^{2}$ = 400.
2
So $2^{20}$ is bigger.
3
The reason: taking logs, $a^{b}$ against $b^{a}$ is the same as $\frac{\ln a}{a}$ against $\frac{\ln b}{b}$ . The function $\frac{\ln x}{x}$ rises until $x = e \approx 2.718$ and falls afterwards. For 2 and 20 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 2^{20} = 1048576\ >\ 20^{2} = 400 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

2^{20}

20^{2}

$2^{20}$ is bigger: $2^{20}$ = 1048576 and $20^{2}$ = 400.

Which is bigger, 2202^{20}220 or 20220^{2}202?

How to work it out

Questions

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Which is bigger, $2^{20}$ or $20^{2}$ ?