Which is bigger, $12^{14}$ or $14^{12}$ ?

Q: Which is bigger, 12^14 or 14^12?

12^14 is bigger: 12^14 = 1.28391 × 10^15 and 14^12 = 5.66939 × 10^13.

$12^{14}$ is bigger: $12^{14}$ = 1.28391 × $10^{15}$ and $14^{12}$ = 5.66939 × $10^{13}$ .

12^{14} = 1.28391\times 10^{15}\ >\ 14^{12} = 56693912375296

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^14 against 14^12 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: $12^{14}$ = 1283918464548864 and $14^{12}$ = 56693912375296.
2
So $12^{14}$ 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 14 are past e, where the function is decreasing, so the larger exponent loses and $12^{14}$ > $14^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 12^{14} = 1.28391\times 10^{15}\ >\ 14^{12} = 56693912375296 \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

12^{14}

14^{12}

$12^{14}$ is bigger: $12^{14}$ = 1.28391 × $10^{15}$ and $14^{12}$ = 5.66939 × $10^{13}$ .

Which is bigger, 121412^{14}1214 or 141214^{12}1412?

How to work it out

Questions

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Which is bigger, $12^{14}$ or $14^{12}$ ?