Which is bigger, $3^{22}$ or $22^{3}$ ?

Q: Which is bigger, 3^22 or 22^3?

3^22 is bigger: 3^22 = 31381059609 and 22^3 = 10648.

$3^{22}$ is bigger: $3^{22}$ = 31381059609 and $22^{3}$ = 10648.

3^{22} = 31381059609\ >\ 22^{3} = 10648

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The curve ln x / x, which decides every comparison of a^b with b^a. 3^22 against 22^3 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: $3^{22}$ = 31381059609 and $22^{3}$ = 10648.
2
So $3^{22}$ 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. Both 3 and 22 are past e, where the function is decreasing, so the larger exponent loses and $3^{22}$ > $22^{3}$ .
$\frac{\ln 3}{3}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 3^{22} = 31381059609\ >\ 22^{3} = 10648 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

3^{22}

22^{3}

$3^{22}$ is bigger: $3^{22}$ = 31381059609 and $22^{3}$ = 10648.

Which is bigger, 3223^{22}322 or 22322^{3}223?

How to work it out

Questions

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Which is bigger, $3^{22}$ or $22^{3}$ ?