Which is bigger, $2^{21}$ or $21^{2}$ ?

Q: Which is bigger, 2^21 or 21^2?

2^21 is bigger: 2^21 = 2097152 and 21^2 = 441.

$2^{21}$ is bigger: $2^{21}$ = 2097152 and $21^{2}$ = 441.

2^{21} = 2097152\ >\ 21^{2} = 441

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^21 against 21^2 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: $2^{21}$ = 2097152 and $21^{2}$ = 441.
2
So $2^{21}$ 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 21 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 2^{21} = 2097152\ >\ 21^{2} = 441 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

2^{21}

21^{2}

$2^{21}$ is bigger: $2^{21}$ = 2097152 and $21^{2}$ = 441.

Which is bigger, 2212^{21}221 or 21221^{2}212?

How to work it out

Questions

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Which is bigger, $2^{21}$ or $21^{2}$ ?