Which is bigger, $13^{22}$ or $22^{13}$ ?

Q: Which is bigger, 13^22 or 22^13?

13^22 is bigger: 13^22 = 3.21183 × 10^24 and 22^13 = 2.82810 × 10^17.

$13^{22}$ is bigger: $13^{22}$ = 3.21183 × $10^{24}$ and $22^{13}$ = 2.82810 × $10^{17}$ .

13^{22} = 3.21183\times 10^{24}\ >\ 22^{13} = 2.82810\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^22 against 22^13 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: $13^{22}$ = 3211838877954855105157369 and $22^{13}$ = 282810057883082752.
2
So $13^{22}$ 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 13 and 22 are past e, where the function is decreasing, so the larger exponent loses and $13^{22}$ > $22^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 13^{22} = 3.21183\times 10^{24}\ >\ 22^{13} = 2.82810\times 10^{17} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

13^{22}

22^{13}

$13^{22}$ is bigger: $13^{22}$ = 3.21183 × $10^{24}$ and $22^{13}$ = 2.82810 × $10^{17}$ .

Which is bigger, 132213^{22}1322 or 221322^{13}2213?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{22}$ or $22^{13}$ ?