Which is bigger, $4^{12}$ or $12^{4}$ ?

Q: Which is bigger, 4^12 or 12^4?

4^12 is bigger: 4^12 = 16777216 and 12^4 = 20736.

$4^{12}$ is bigger: $4^{12}$ = 16777216 and $12^{4}$ = 20736.

4^{12} = 16777216\ >\ 12^{4} = 20736

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^12 against 12^4 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: $4^{12}$ = 16777216 and $12^{4}$ = 20736.
2
So $4^{12}$ 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 4 and 12 are past e, where the function is decreasing, so the larger exponent loses and $4^{12}$ > $12^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 12}{12}$

LaTeX source of the formulas on this page

\[ 4^{12} = 16777216\ >\ 12^{4} = 20736 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 12}{12} \]

Questions

Which is bigger,

4^{12}

12^{4}

$4^{12}$ is bigger: $4^{12}$ = 16777216 and $12^{4}$ = 20736.

Which is bigger, 4124^{12}412 or 12412^{4}124?

How to work it out

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Which is bigger, $4^{12}$ or $12^{4}$ ?