Which is bigger, $12^{22}$ or $22^{12}$ ?

Q: Which is bigger, 12^22 or 22^12?

12^22 is bigger: 12^22 = 5.52061 × 10^23 and 22^12 = 1.28550 × 10^16.

$12^{22}$ is bigger: $12^{22}$ = 5.52061 × $10^{23}$ and $22^{12}$ = 1.28550 × $10^{16}$ .

12^{22} = 5.52061\times 10^{23}\ >\ 22^{12} = 1.28550\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^22 against 22^12 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: $12^{22}$ = 552061438912436417593344 and $22^{12}$ = 12855002631049216.
2
So $12^{22}$ 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 12 and 22 are past e, where the function is decreasing, so the larger exponent loses and $12^{22}$ > $22^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 12^{22} = 5.52061\times 10^{23}\ >\ 22^{12} = 1.28550\times 10^{16} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

12^{22}

22^{12}

$12^{22}$ is bigger: $12^{22}$ = 5.52061 × $10^{23}$ and $22^{12}$ = 1.28550 × $10^{16}$ .

Which is bigger, 122212^{22}1222 or 221222^{12}2212?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{22}$ or $22^{12}$ ?