Which is bigger, $12^{13}$ or $13^{12}$ ?

Q: Which is bigger, 12^13 or 13^12?

12^13 is bigger: 12^13 = 1.06993 × 10^14 and 13^12 = 2.32980 × 10^13.

$12^{13}$ is bigger: $12^{13}$ = 1.06993 × $10^{14}$ and $13^{12}$ = 2.32980 × $10^{13}$ .

12^{13} = 1.06993\times 10^{14}\ >\ 13^{12} = 23298085122481

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^13 against 13^12 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: $12^{13}$ = 106993205379072 and $13^{12}$ = 23298085122481.
2
So $12^{13}$ 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 13 are past e, where the function is decreasing, so the larger exponent loses and $12^{13}$ > $13^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 12^{13} = 1.06993\times 10^{14}\ >\ 13^{12} = 23298085122481 \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

12^{13}

13^{12}

$12^{13}$ is bigger: $12^{13}$ = 1.06993 × $10^{14}$ and $13^{12}$ = 2.32980 × $10^{13}$ .

Which is bigger, 121312^{13}1213 or 131213^{12}1312?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{13}$ or $13^{12}$ ?