Which is bigger, $2^{22}$ or $22^{2}$ ?

Q: Which is bigger, 2^22 or 22^2?

2^22 is bigger: 2^22 = 4194304 and 22^2 = 484.

$2^{22}$ is bigger: $2^{22}$ = 4194304 and $22^{2}$ = 484.

2^{22} = 4194304\ >\ 22^{2} = 484

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^22 against 22^2 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: $2^{22}$ = 4194304 and $22^{2}$ = 484.
2
So $2^{22}$ 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 22 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 2^{22} = 4194304\ >\ 22^{2} = 484 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

2^{22}

22^{2}

$2^{22}$ is bigger: $2^{22}$ = 4194304 and $22^{2}$ = 484.

Which is bigger, 2222^{22}222 or 22222^{2}222?

How to work it out

Questions

Similar comparisons

Which is bigger, $2^{22}$ or $22^{2}$ ?