Which is bigger, $12^{28}$ or $28^{12}$ ?

Q: Which is bigger, 12^28 or 28^12?

12^28 is bigger: 12^28 = 1.64844 × 10^30 and 28^12 = 2.32218 × 10^17.

$12^{28}$ is bigger: $12^{28}$ = 1.64844 × $10^{30}$ and $28^{12}$ = 2.32218 × $10^{17}$ .

12^{28} = 1.64844\times 10^{30}\ >\ 28^{12} = 2.32218\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^28 against 28^12 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: $12^{28}$ = 1648446623609512543951043690496 and $28^{12}$ = 232218265089212416.
2
So $12^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $12^{28}$ > $28^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 12^{28} = 1.64844\times 10^{30}\ >\ 28^{12} = 2.32218\times 10^{17} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

12^{28}

28^{12}

$12^{28}$ is bigger: $12^{28}$ = 1.64844 × $10^{30}$ and $28^{12}$ = 2.32218 × $10^{17}$ .

Which is bigger, 122812^{28}1228 or 281228^{12}2812?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{28}$ or $28^{12}$ ?