Which is bigger, $6^{12}$ or $12^{6}$ ?

Q: Which is bigger, 6^12 or 12^6?

6^12 is bigger: 6^12 = 2176782336 and 12^6 = 2985984.

$6^{12}$ is bigger: $6^{12}$ = 2176782336 and $12^{6}$ = 2985984.

6^{12} = 2176782336\ >\ 12^{6} = 2985984

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^12 against 12^6 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: $6^{12}$ = 2176782336 and $12^{6}$ = 2985984.
2
So $6^{12}$ 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 6 and 12 are past e, where the function is decreasing, so the larger exponent loses and $6^{12}$ > $12^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 12}{12}$

LaTeX source of the formulas on this page

\[ 6^{12} = 2176782336\ >\ 12^{6} = 2985984 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 12}{12} \]

Questions

Which is bigger,

6^{12}

12^{6}

$6^{12}$ is bigger: $6^{12}$ = 2176782336 and $12^{6}$ = 2985984.

Which is bigger, 6126^{12}612 or 12612^{6}126?

How to work it out

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Which is bigger, $6^{12}$ or $12^{6}$ ?