Which is bigger, $8^{12}$ or $12^{8}$ ?

Q: Which is bigger, 8^12 or 12^8?

8^12 is bigger: 8^12 = 68719476736 and 12^8 = 429981696.

$8^{12}$ is bigger: $8^{12}$ = 68719476736 and $12^{8}$ = 429981696.

8^{12} = 68719476736\ >\ 12^{8} = 429981696

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^12 against 12^8 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: $8^{12}$ = 68719476736 and $12^{8}$ = 429981696.
2
So $8^{12}$ 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 8 and 12 are past e, where the function is decreasing, so the larger exponent loses and $8^{12}$ > $12^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 12}{12}$

LaTeX source of the formulas on this page

\[ 8^{12} = 68719476736\ >\ 12^{8} = 429981696 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 12}{12} \]

Questions

Which is bigger,

8^{12}

12^{8}

$8^{12}$ is bigger: $8^{12}$ = 68719476736 and $12^{8}$ = 429981696.

Which is bigger, 8128^{12}812 or 12812^{8}128?

How to work it out

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Which is bigger, $8^{12}$ or $12^{8}$ ?