Which is bigger, $22^{30}$ or $30^{22}$ ?

Q: Which is bigger, 22^30 or 30^22?

22^30 is bigger: 22^30 = 1.87361 × 10^40 and 30^22 = 3.13810 × 10^32.

$22^{30}$ is bigger: $22^{30}$ = 1.87361 × $10^{40}$ and $30^{22}$ = 3.13810 × $10^{32}$ .

22^{30} = 1.87361\times 10^{40}\ >\ 30^{22} = 3.13810\times 10^{32}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^30 against 30^22 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: $22^{30}$ = 18736153019903829443036278993864332673024 and $30^{22}$ = 313810596090000000000000000000000.
2
So $22^{30}$ 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 22 and 30 are past e, where the function is decreasing, so the larger exponent loses and $22^{30}$ > $30^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 22^{30} = 1.87361\times 10^{40}\ >\ 30^{22} = 3.13810\times 10^{32} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

22^{30}

30^{22}

$22^{30}$ is bigger: $22^{30}$ = 1.87361 × $10^{40}$ and $30^{22}$ = 3.13810 × $10^{32}$ .

Which is bigger, 223022^{30}2230 or 302230^{22}3022?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{30}$ or $30^{22}$ ?