Which is bigger, $6^{13}$ or $13^{6}$ ?

Q: Which is bigger, 6^13 or 13^6?

6^13 is bigger: 6^13 = 13060694016 and 13^6 = 4826809.

$6^{13}$ is bigger: $6^{13}$ = 13060694016 and $13^{6}$ = 4826809.

6^{13} = 13060694016\ >\ 13^{6} = 4826809

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^13 against 13^6 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: $6^{13}$ = 13060694016 and $13^{6}$ = 4826809.
2
So $6^{13}$ 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 6 and 13 are past e, where the function is decreasing, so the larger exponent loses and $6^{13}$ > $13^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 6^{13} = 13060694016\ >\ 13^{6} = 4826809 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

6^{13}

13^{6}

$6^{13}$ is bigger: $6^{13}$ = 13060694016 and $13^{6}$ = 4826809.

Which is bigger, 6136^{13}613 or 13613^{6}136?

How to work it out

Questions

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Which is bigger, $6^{13}$ or $13^{6}$ ?